FPS (Single Player): STALKER Call of Pripyat

FPS (Multiplayer): UT2004

RPG: I'm kinda torn here... Planescape and New Vegas are both strong contenders.

MMORPG: n/a

Action/Hack-n-Slash: Far too vast to pick just one...

Fighting: Zeno Clash, if that counts. I think it does.

Racing: GRID. A lot of strong contenders, but that one just shines.

RTS: Supreme Commander (Forged Alliance). FA for multiplayer and skirmish. The original for the campaign. The sequel is alright for a change of pace but really does oversimplify.

Adventure/Point-and-Click: Another tough one. I have to agree that Machinarium has so much to recommend it. Not the longest game ever. But so beautifully constructed.

Casual: Minecraft.

Management/Simulation: Tropico 4.

TBS: X-Com Apocalypse. I know real time is an option. But it's an option I ignored.

Grand Strategy: Medieval 2 Total War. Shogun 2 may be better but I have invested less time in that one so far so I can't really comment.

FPS (Single Player): BLOOD - I've been obsessed with this game since it's release. I'm a big horror movie fan, so it's only natural that I'd dig this game.

FPS (Multiplayer): Doom II - Barely edges out Golden Eye for the simple fact that I was fortunate enough to play Doom II on a LAN setup vs like 20 other kids in my computer science class in high school.

RPG: Fallout 2 - The mature nature and depth really blew me away at the time of it's release.

MMORPG: World of Warcraft, I guess. MMO's suck though. The last one I played was Age of Conan, really just a WoW clone though.

Action/Hack-n-Slash: Diablo by a mile. One of my alltime favorite games.

Fighting: Super Street Fighter IV. My alltime favorite game, edges out Street Fighter II Turbo because of the mostly lag-free online multiplayer. Being able to fight somebody anytime you want is crack.

Racing: I guess Super Mario Kart, other than that, I have no interest in racing games.

RTS: Starcraft, but after a while it felt more like work than fun.

Adventure/Point-and-Click: Shadowgate, mostly for the atmosphere and nostalgia.

Casual: Super Puzzle Fighter II Turbo I guess?

Other (Anything that doesn't comfortably fit in any above category): Roguelikes are missing.... Dungeon Crawl Stone Soup is one of the greatest games I've ever played.

Turn Based Strategy - Jagged Alliance 2 - a real gem that I luckily stumbled upon back in the day.

FPS (Single Player): Blood (Excellent horror theme, fun weapons, fun enemies, fun levels and fun gameplay)
FPS (Multiplayer): Quake 3 (though I only play with bots, it is fast and excellent at what it does)
RPG: Mount and Blade: Warband (especially the Storm of the Three Kingdoms mod)
MMORPG: Don't really play, but I enjoyed the trial of Age of Conan (pity it's prettier now and won't run on my computer)
Action/Hack-n-Slash: Diablo or Drakengard (Diablo has far better atmosphere than its sequel, and I prefer the gameplay to 2, and Drakengard is a damn fun hack and slash game with a grimdark storyline)
Fighting: Darkstalkers (Street Fighter 2, with horror monsters!)
Racing: Don't play racing games
RTS: Myth 2 (Brutally hard, but so much fun, and it is so satisfying to have your dwarves blow up and entire aquad of enemies)
TBS: X-Com (challenging, simple to learn, and just damn fun)
Adventure/Point-and-Click: Alone in the Dark (I love Alone in the Dark. I love the atmosphere, the feel, the only problem I have with it is the double tap to run thing)
Casual: I dunno
Other: System Shock 2 and Princess Maker 2 (System Shock 2 is beautiful, bringint ogether horror, sci-fi, FPS and RPG together perfectly. Everytime I play Deus Ex, I can't help but think, "I wish this was more like System Shock 2." Princess Maker 2 is just lots of fun)

FPS (Single Player): I don't really play this genre but I did enjoy Unreal Tournament 2004.
FPS (Multiplayer): Haven't played any yet but I'm planning to rock some ** with TF2
RPG: Baldur's Gate. There are several contenders for this position but this one hits the sweet spot for me. Might and Magic 7 is great too.
MMORPG: I've probably most enjoyed FF XIV alpha/beta.
Action/Hack-n-Slash: Kingdom Hearts hands down. Diablo II was cool, but I can play KH for long periods of time. On the NDS Tales of Innocence and Soma Bringer are very good.
Fighting: *dooms* Soul Calibur 3.
Racing: Hmm...I've enjoyed...oh! I know! The log racing mini-game in Croc 2 : Legend of Gobbos.
RTS: HOMAM II! Not Gold, but the normal version without a datadisc. Oh wait, you meant real time? Then my pick is Age of Empires II.
Adventure/Point-and-Click: Quest for Glory 1 and 3 and King's Quest VI all the way! And they're all nestling on the same branch...well, they're all great. Besides,QfG is partially an RPG too, so if we went for a pure adventure, KQVI is the best adventure ever. For me.
Casual: Trine. Or what do you define as casual? Anyhow, I love Trine. TadaDAdaDAda.
Open world: Morrowind. Great atmosphere, I loved the magic system.
Other: Golden Sun gets an honorable mention. If you have a GBA, you owe it to yourself to play it - it has a really moving, atmospheric plot that's emphasized by wonderful music and equally great graphics. And the RPG system is very good and doesn't get boring. The best RPG for handhelds that I've ever come across. And no, I prefer FF VIII to the other FF sequels, I'm not a big fan of FF VI or IV (FF8 is elbowing on my best RPG first position together with BG and GS). Hehe. I should install FF8 again and see how much I like it now.
People think it's for kids when it's not: Pokemon. Anyone who's played them as a child knows what I'm talking about. For kids or not, it's a great series and not only for that 20+ year olds are able to play it and have a roaring good time even today.

Interesting idea. :) Here are my picks:

FPS (Single Player): Prey. Surreal surroundings and several interesting concepts (ghost form, gravity changes), though not all of them worked out. The story ended up being a cookie-cutter "guy saves his girl" plot despite the promising beginning ... but kind of lost hope for engaging storylines in FPS games anyway.

FPS (Multiplayer): none. Never played one. Not my cup of tea.

RPG: Morrowind. Unique atmosphere, epic plots, huge size, enhanced by thousands of mods.

MMORPG: Guild Wars. No monthly fees. Fully instanced, and fully playable with AI-controlled heroes, so I can play it like a single player game. Nice concept of skills which enable hundreds of different character builds, and encourages experimenting.

Action/Hack-n-Slash: Not sure what this category is supposed to be My favorite Hack'n'Slash game is probably NetHack, but i doubt that that's what you had in mind.

Fighting: none. The most positive thing that I can say about any fighting game I tried is that it took a full 20 minutes before it started to either bore or disgust me.

Racing: Crazy Cars 3. Very old game, but the formula worked well, and I played it a lot.

TBS: Civilization IV. Deep, complex, good AI, and has tons of mods. Though it really hurts to have to pick only one, this is "my" favorite genre and there are half a dozen other games that would at least deserve a mention.

RTS: Europa Universalis III. Deep, complex, lots of content, and I like the historical setting, especially the fact that it simulates a whole "world" with dozens of nations instead of just a handful of "players".

Adventure/Point-and-Click: Indiana Jones and the Fate of Atlantis. professionally done characters and alternative storylines. Unfortunately there's no category for text adventures (which I prefer over the point-and-click ones); in that category, "Suspended" would be my pick.

Casual: Bookworm. One of the few casual games that actually captured my interest for a while. Or "Tetris", if you count that as casual, but it's debatable because it was released when the term "casual games" hadn't even been coined yet.

Other (Anything that doesn't comfortably fit in any above category): Star Control II Unique mix of adventure and space fight that defies any categorization. :)

FPS (Single Player): Not my genre, really. Doom is one that I first played for well over fifteen years ago and I still return to it occasionally. The soundtrack kicks ass, too.
EDIT: I can't believe how I forgot about F.E.A.R. here. The levels are a bit repetitive and tedious, but the firefights are intense. And what other games let you staple a guy into the ceiling from his ass?

FPS (Multiplayer): GoldenEye and the PC version of Quake II; as I said, I rarely play FPS's, and these two games happen to be the only ones I can remember playing with friends.

RPG: Even though I like the Baldur's gate series and Morrowind more than I like most people, they can't hold a candle to Planescape: Torment - the best game ever made and one that everyone should experience before they die. Because of its numerous little foibles, I know that Torment isn't a perfect game, and yet... it just is. Every time I play Torment, I'm so engrossed in it that if my brother suddenly burst into my apartment on fire and in terrible agony, I wouldn't pay him enough attention to tell him to bugger off.

Fighting: Super Smash Brothers: Melee. It's easy to pick up yet difficult to master. It's incredibly chaotic and extremely hectic. I have played Tekkens, Soul Caliburs, Mortal Kombats and whatnot as well, but the ridiculously difficult moves and comboes are too off-putting for me.

Action: The console version of Resident Evil 4. The plot is utter drivel and the game itself doesn't feel Resident Evil-ish at all, but I'll be damned if I care when there are a thousand enemies to bury deep.

Racing:While I do enjoy Need For Speeds and such arcade-ish racing games, they have failed to impress me like Gran Turismo 4. What can I say, really? There are over 700 cars to choose from (even if there are what, 50 variations of the Nissan Skyline?) and the realism is pretty damn convincing. The only game that has made me venture into the realms of the Internet in search for an affordable wheel controller.
EDIT: Also, there's F-Zero GX. Sometimes it feels more challenging than life itself, but it just makes victory all the sweeter, even if you've crossed the finish line neck-to-neck with another racer and your heart is beating a thousand times a minute. The only game where I have bothered with trying to attempt the fastest possible lap in Time Attack. I even made a custom vehicle that's terrifyingly fast when it's on the right mood and a real bitch of a handler when it isn't.
Oh, and there's a brilliant soundtrack, too.

Hmm...

FPS (SP): Not DOOM. Nope.
FPS (MP): Hard choice, ultimately I'll say Quake III. The definitive DM experience.
RPG: The Witcher 2 is my new favorite.
TBS: Hard choice between X-Com and Jagged Alliance 2. Honestly can't choose.
RTS: Command & Conquer: Red Alert 2 with Yuri's Revenge expansion pack.
MMORPG: -NOPE-
Action/H&S: Diablo II springs to mind.
Fighting: Mortal Kombat II.
Racing: MOTHERFUCKIN' WACKY WHEELS.
Adventure/Point-and-Click: Sam & Max Hit the Road.
Open World: Grand Theft Auto III.
Casual: Tetris.
Other: It appears you are missing Simulators, so I'll list my favorite Space Sim: Wing Commander.

EDIT: You also forgot Side Scrollers! Commander Keen is sad. :C

FPS (Single Player): System Shock or Deus Ex
RPG: Baldur's Gate 2
Action/Hack-n-Slash: Diablo 2
TBS: Age of Wonders: Shadow Magic
Adventure/Point-and-Click: I gave up on Adventure games 20 years ago, but I really liked Legend of the Sword.
Other: Thief - The Dark Project

FPS (Single Player): Half Life 1
FPS (Multiplayer): Red Orchestra or UT1
RPG: Fallout 2
MMORPG: none
Action/Hack-n-Slash: Nox
Fighting: MK4
Racing: NFS Underground (but i don't really like racing)
RTS: Red Alert 1
Adventure/Point-and-Click: TLJ
Casual: armorgames.com :)
Other (Anything that doesn't comfortably fit in any above category): Sandbox: Terraria.

FPS (Single Player): Half-Life 2. I find the gameplay extremely fun and the story is great.

FPS (Multiplayer): Left 4 Dead 2. This one is actually kind of a lie; I never play multiplayer. But if I did, L4D2 would win. The gameplay is fun, the levels are great, and the enemies are awesome. Zoey will always be my favorite character, though.

RPG: Knights of the Old Republic I or II. I love the stories, combat, and characters from both, so I can't pick just one of them. Mass Effects 1 and 2 are runners-up. And if JRPGs are included, my favorite is Final Fantasy XIII.

TBS: Elven Legacy. It's actually the only TBS game I own, so it can't not win. I haven't played it much, though, so...

MMORPG: Have never played one, nor do I plan on doing so.

Action/Hack-n-Slash: Sacred Gold. It's the only game I can think of that would fit this category. At any rate, I find the game fun. The combat's fun and the story isn't too shabby.

Fighting: Street Fighter IV. I haven't played many fighting games, but I've spent quite a bit of time with this one, so it wins. The fighting looks cool and the almost nonsensical story can be amusing.

Racing: Project Gotham Racing 4. A terrific game, through and through. The driving is tight and awesome. There's nothing quite like the feel of performing a perfect drift.

RTS: Dawn of War. Company of Heroes almost won, but Dawn of War has a better story. The gameplay is nicely balanced and the graphics are great for a seven-year-old game.

Adventure/Point-and-Click: Escape from Monkey Island. It's not point-and-click, but it is an adventure game. I have many memories from this one. The characters are okay and the humor is excellent. The story is pretty interesting, too. Monkey Combat stinks, though.

Open World: The Elder Scrolls IV: Oblivion. As much as it is criticized, I love the game. The combat is fun, the story isn't too bad, and the sheer amount of stuff to do is incredible. The game isn't perfect, but I like it anyways.

Casual: Minesweeper. Okay, so it comes with Windows. But seriously, it's awesome.

Other: Alone in the Dark (2008). Occasionally a buggy mess, I love this game despite its faults. The story is interesting (even if the ending is abrupt and doesn't explain much) and the gameplay is unique. I even liked the jacket/inventory.

FPS (Single Player): Half Life 2 - The action's great and the in game universe is interesting.

FPS (Multiplayer): Team Fortress 2 - I've probably never had as much fun playing a game and it has provided me with hundreds of hours of entertainment.

RPG: Kotor 2 - In most rpg's i play every decision has a very obvious alignment and obvious consequences. Kotor 2 not only made me think about my choices in the game but also outside of it. The melancholic atmosphere and characters with flaws who made me give a damn are part of what makes this my favourite game of all time.

MMORPG: Eve Online - I love the meta-gaming and the amount of freedom they give you.The interactions with other players are the most important aspect as it should in an mmo.

Action/Hack-n-Slash: VVVVVV - Manages to infuriate the hell out of me one minute and charm me the next.

Fighting: Killer instinct 2 - The fighting game I've played the most and the only one where i managed to learn the best moves.

Racing: Mario-Kart 64 - I find racing games boring but i used to love playing this.

RTS: Total Annihilation - First game i was ever really involved in and was responsible for my love of gaming ever since. Bar the pathfinding it hasn't really been improved upon in its genre.

Adventure/Point-and-Click: The Longest Journey - I'm not a fan of adventure games but i still enjoyed playing it.

Casual: Tetris - Manages to still be entertaining even though Ive played it all of my life.

truth. Minesweeper is awesome.

FPS (Single Player): Half-Life - I keep trying to come up with a more clever answer but I keep coming back to the blast chamber level...
FPS (Multiplayer): I don't play multi-player FPSes multiplayer but I enjoyed the heck out of Unreal Torney against the computer, back in the dizzle
RPG: Ooh it's got to be Infinity Engine but is it Baldur's Gate 2 or Planescape Torment? I'm going with BG2 for a more rounded gaming experience - good story (not AS good but good) with better combat - actual ranged weapons and everything!
MMORPG: N/A
Action/Hack-n-Slash: If you mean 'Action RPG' I'm going to include RGP/FPS hybrid and say Deus Ex, if you don't like me doing that you can take Nox instead
Fighting: Soul Caliber 2 on XBox
Racing: As much as I've played Project Gotham 2 et al in recent years, I'm going to go with the original Microprose GP on the Amiga - that thing had me tweaking my car setups for specific circuits and saving them off to 3.5" floppy and everything!
Adventure/Point-and-Click: Broken Sword 1 - the cinematics of that game alone blew me away and I haven't even played the HD re-make yet!
Open World: Mercenary on the C64 - there, you weren't expecting that!
Casual: I saw a remake of BouderDash classed as 'casual' the other day so I'll say the C64 version
Other (Anything that doesn't comfortably fit in any above category): Outcast!
No-one has said that you missed flight, I would have said original Wing Commander on Amiga!

FPS (Single Player): Half Life 2. I pretty much liked the story and presentation, silent Gordon notwithstanding. Good action and supporting characters and not just "shoot at everything that moves".
FPS (Multiplayer): Unreal Tournament 2004. Fast and fun, pretty and I could play it to destress myself quite nicely.
RPG: Baldur's Gate 2 was quite enjoyable and it really mixes strategy with your role playing.
TBS: Haven't played that many turn based games and never finished one. It's on the "to-do" list.
MMORPG: No intention of ever playing a paying MMORPG and I've only dabbled for a couple of missions on free ones.
Action/Hack-n-Slash: I actually enjoyed Nox very much. Loved the way arrows flew and the humor in some spots.
Fighting: Not a fan of the genre.
Racing: Not a fan of the genre.
Platformer: Not a fan, though I enjoyed and finished Psychonauts, mostly due to the presentation and humor, rather than it's platformy nature.
RTS: Starcraft. Both versions, still playing the second, usually against friends.
Adventure/Point-and-Click: Myst is the only one I've ever finished. I've never bothered to spend too much money on adventure games due to their non-replayability, though I've got a few waiting on the backlog and quite enjoy figuring out puzzles.
Casual: For some reason, I've a thing for Droplitz. Unexplainable, really.
Other (Anything that doesn't comfortably fit in any above category): You pretty much covered everything.

I've been playing Need for Speed online with my brother, even though I'm not a fan of racing games, but I've noticed that I suck at having "control skills". I suck at games that demand pressing a combination of buttons in a specific instant, which you need in racing, fighting and platformer games. This makes me not good at Starcraft, but neither are my friends, so we enjoy it at a low skill level.

FPS (Single Player): Halo Series, sure, the later games are short, but at least zjey are a real challenge on Legendary setting.
FPS (Multiplayer): Golden Eye (N64), everyone who played this gem in 4 Player Split Screen back in the day knows why, it´s just pure fun, most of the maps where great and it fas pure fun.
RPG: Baldur´s Gate, the first one, got me into all that Fantasy stuff.
TBS: Heroes of Might & Magic 3, we always played that during LAN Partys, had it open for the entire duration of the Party and during Mapchanges or short breaks in the normal games we made a few turns.
MMORPG: Auto Assault, sadly it´s offline for years now, but it was Endless fun.
Action/Hack-n-Slash: Sacred 2. Nuff said.
Fighting: Super Street Fighter II (SNES), it was the version we always played back in the day and we always had endless fun, even that we never figured out all the Moves (never had a manual) or where any good at it
Racing: Grand Prix Legends (BTW, this wood be an awesome adition for GOG), love this game, best freaking simulation ever!
Platformer: Sonic the Hedgehog (Mega Drive / Genesis), kicked Marios Ass seriously, awesome game, play it every year at least once.
RTS: Total War Series, but I like Shogun 2 the most.
Adventure/Point-and-Click: Day of the Tentacle, nuff said
Casual: Peggle
Other (Anything that doesn't comfortably fit in any above category): Beathazzard Ultra, best Music based Game ever.