Because we game buyers then complain that the developer is milking the franchise - with the same characters and game mechanics - for a cash grab.

Sometimes it's simply because the developers change publishers and lose the rights to the name. System Shock became BioShock which is mostly designed by the same people and probably would have been a System Shock game otherwise.

Half Life is probably stuck at 2 because Valve are making far too much money from Steam that they don't even remember making games anymore. They'd have to farm it out to a contractor and they're worried they'd fuck it up.

Otherwise i think most publishers are more than happy to milk a series until it's dead and then flog it a bit to make sure.

I think that is more of a special case, because the real SMB 2 was the "stinker" (or in this case the exact same game with harder levels that weren't casual friendly), and not even worth an international releases. Thus the "star" releases were 1 and 3. This is another phenomenon that happens frequently in games as well. The most common scenario is: hit game makes a sequel and changes it up to be fresh and interesting. Everyone thinks it's lame and wonders why they changed it so much, or in a certain way, then part 3 goes back to being more like the original.

Sure, with movies, it rarely seems to be great, but with video games, I so often find that the third game is where they really hit their stride.

Super Mario Bros. 3
Onimusha 3
Mega Man 3
Metroid 3: Super Metroid
The Legend of Zelda: A Link to the Past
Heroes of Might and Magic III
Command and Conquer III: Tiberium Wars (granted, if we're counting spinoffs, they hit their stride back with RA2)
WarCraft III: Reign of Chaos

Precisely. There are lots of game series where my favourite is #2.

Its regarded as good to stop at 2, not just in gaming. Just look at Fawlty Towers, The Office. But yes, there does seem to be some inherent need, especially in Fantasy novels, to have three. Assassins Quest by Russel Hobb is another example, the 3rd book reads like it was written by a different person. However to counter that argument, HL2 was pretty poor, more of a "looky here we have a gravity gun" simulator.

I don't expect there is a single source for the phenomen, I mean fantasy novels, yes probably saw the success of LOtR and follow that. But there have been many other significant numbers,
http://mysticalnumbers.com/number-3
7 deadly sins
7th son of a 7th son
etc.

Star Wars Battlefront by EA.

MONEY

Many 3rd games sucked, besides the ones mentioned already, like Lords of the Realm 3, Master of Orion 3, Gothic 3 - Forsaken gods (though technically a standalone expansion to Gothic 3).

Megaman X 3
Megaman Zero 3

There's something about Megaman and 3 that's a winning combination.

And don't forget Leg- Oh... *Goes off and cries.*

And this : http://www.rockmanpm.com/i/flash/rmzxaf.html
(it's an April Fool from Capcom, unfortunately)

Damn, no.3 really is cursed.

This. I think it's a symptom of cycles of testing. First game is the proof of concept -- is it going to succeed? Where did we go wrong, how can we fix it? If First succeeds, a doorway of potential is opened... Second takes all the mistakes and lessons of First, fixes and polishes them. Second is essentially the First, but matured and fully realized, at the height of its power.

If Second also succeeds, Third continues what was good in Second, but probably isn't as ground-breaking -- after all, the devs made all the mistakes with First, succeeded with Second, and want to ride on that wave of success with Third and subsequent games. So Third has potential to become the gateway to milking the market for all it's worth, keeping that door of potential open as long as possible... long after it was worthwhile. If Third won't wrap up the trilogy properly, then you get that downhill slide.

Assassin's Creed is the prime example. Maybe the same happened with Halo, Fallout, Bioshock, and Final Fantasy XIII. Baldur's Gate, Mass Effect and The Witcher ended at the right place (then again I have a good opinion of ME3). Sometimes the trilogy just needs to end, right here, for good.

Still waiting on The Darkness 3. And possibly a port of the first one. That's a really beautiful game series.

Because the sequel has to be bigger in some way to the previous game and the game maker loses sight on what made the first game good or popular.

Same could be said with movies.