So many games on this console!

The only one I know and actually have completed several times via an emulated version:
Shadow of the Colossus

I think I've barely even played much more than a dozen or so, but from what I've seen my favorites were

1. Jak and Daxter: The Precursor Legacy (by a wide margin)
2. Okami (now also on PC)
3. ICO
4. The Hobbit (also on PC, but sadly no digital version anywhere)
5. Legend of Kay (also on PC)
6. Asterix and Obelix XXL - or was it XXL 2? (also on PC)
7. Baldur's Gate: Dark Alliance
8. Bully / Canis Canem Edit
9. Champions of Norrath
10. God of War

(God of War is so low on the list because I didn't really get far in it, just wasn't quite my type of thing, although I can see the appeal.)

1. Xenosaga
2. Xenosaga 2
3. Xenosaga 3
4. Dragon Quest 8
5. FF 10
6. Mortal Kombat Shaolin Monks -it reminds me of double dragon games-
7. GTA Vice City -why get the remaster? lol-
8. GTA San Andreas
9. Legends of Wrestling 2
10. Ace Combat 5

1. Shadow Hearts: Covenant
2. Shadow Hearts: From The New World
3. Devil May Cry 3
4. Dragon Quest 8
5. Persona 4
6. Persona 3
7. Onimusha: Dawn of Dreams
8. Summoner 2
9. God of War 2
10.Devil May Cry 2

Mostly JRPGs and Hack & Slay games.

1. Street Fighter Alpha Anthology
2. Tony Hawk's Underground 2
3. Star Wars Episode III: Revenge of the Sith
4. Tekken 4
5. Tomb Raider: Angel of Darkness
6. Virtua Fighter 4
7. Legacy of Kain - Blood Omen 2
8. Spider-Man 2
9. Backyard Wrestling 2: There Goes The Neighborhood
10. Final Fight Streetwise

No particular order:

1. Final Fantasy X
2. Radiata Stories
3. Klonoa 2
4. Onimusha (first one, but the second one could easily take this spot)
5. Sould Calibur III
6. Tekken 5 *
7. Red Faction
8. God of War**
9. Katamari Damacy
10. SOCOM 3 US Navy Seals***

*(Worth mentioning that Virtua Fighter 4 is a fantastic fighting game as well.)
** God of War II is a very close game to the first one, however everything about the setting of the first game was handled better than the second. Contrary to people, either being ironic or not, insisting these games have good stories I find them to be basically dumb. However, since the first one is dumb in that campy monster movie sort of way I like it regardless.
***I really like these games actually since they remind me of watching Modern Marvels and Future Weapons on History and Discovery Channel respectively. The second game is good and I enjoy the first one even. However, since I do not think I have ever beaten them (well, many of the other games here are like that as well) I opted toward this one.

Last thing was that Genji Dawn of the Samurai is just behind both of the Onimusha games with the only issue it has being that the final boss is so cheap and impossible (at least for me) that I have not beaten it. What's more, all of my actual PS2s have become finicky so I only play on a PS3 anymore and I would have to play the game over again.

Edit: I somehow forgot Tekken Tag Tournament, which is an excellent game... I like it slightly more than Tekken 5 but if I had to pick I'd still go with the latter.

Why does everyone associate these GTA games with PS2? They came to PC just a couple of months later.

Anyway, I would go with the MGS games, though I played them on PS3.

Never had this console, except for the MGS games it's not worth mentioning, just an intermediary between PS1 and the ultimate PlayStation in PS3.

Probably because it's what they had them on. I never had the games back in the day, but I had older cousins who did. One of them had a PC more than capable of playing the GTA games (he got Doom 3 on launch), but he played them on PS2 (Actually San Andreas may have been the Xbox version...hard to remember now.).

My top 10 in no particular order...

Silent Hill 2
Grandia III
Dragon Quest VIII
Lord of the Rings The Third Age
Shadow Hearts
Final Fantasy X
Gran Turismo 3
NCAA Football 2010
WWE Smackdown vs Raw 2008
God of War

1. Gran Turismo 3
2. Metal Gear Solid 2
3. Silent Hill 2
4. God of War
5. Burnout 3: Takedown
6. Virtua Fighter 4 Evolution
7. Tekken 5
8. Gran Turismo 4
9. Ridge Racer V
10. Ace Combat 4

GT2 was a PS1 game, GT3 and 4 were in PS2.
I played the heck out of GT2 and 3 and in my opinion, maybe the 3 most influential racing games ever (with Colin McRae 2.0).

I didn't had a PS2 (PS1 -> PC) but my brother had one, the games I remember foundly are:
Gran Turismo 3
Buzz - party game with dedicated controller. Oh the good times, two "adults" playing Buzz with lots of beer...
Tenchu

Edit:
Almost forgot all the hours on Tourist Trophy trying to beat big bikes on a small TZ125 Yamaha.
Riding Spirits on a friends home, we beat that game several times. Was not that good and probably plays like crap nowadays but takes skill to ride fast and not crash. Was with this game that I learnt the PS2 controller had analog buttons, both the D-Pad and XOSquareTriangle were pressure sensitive.
Edit:/

I played Metal Gear 2 on the PC years later (2010-2012?) and liked the game a lot but I do understand people being frustated with not being able to play with Solid Snake at the time.

A couple of years ago played Shadow of the Colossus on PCSX2 and despite the flaws, the atmosfere is great and battles feel intense, maybe due the contrast of gameplay.

A BIG honorable mention has to go to Richard Burns Rally, it plays like crap on the PS2 but is still one one the better games on the genre, even today without mods.

I don't think i've completed that many PS2 games... But let's see...

1) Odin Sphere
2) FF 12
3) Bloodrayne
4) FF X
5) FF X-2
6) Armored Core 3 (Haven't beat)
7) Disgaea 2
8) Disgaea 1
9) Rogue Galaxy (Haven't beat)
10) Kingdom Hearts (Haven't beat)

None? As weird as it may sound, I've never actually played a single PS2 game. Hell, I can't even recall if I ever tried any of their demo stations back in the day.

I've been thinking of giving some PS2 emulation a try, considering as it's supposed to be pretty damn robust these days with virtually everything being perfectly playable (minus some edge cases).

Hard question, since I enjoyed most games I played, and I don't usually sort games by favoritism "<.<

But I guess the ones I enjoyed the most, sorted alphabetically, and not by favoritism, are:

Devil May Cry 1
Devil May Cry 3
Dragon Ball Budokai Tenkaichi 2
Final Fantasy X
Final Fantasy XII
GrimGrimoire
Naruto Ultimate Ninja 5
Okami
Persona 3 FES
The Simpsons Game