Fallout 3 is one of the best games.

If you like Diablo like games, then maybe Sacred 2.

Burnout paradise if you like action racing.

Grand Theft Auto IV for strong story with action.

For a wonderful storyline and some heart-stopping moments, I can thoroughly recommend Bioshock. Dirt cheap, too :)

I'm also a big fan of Resident Evil 5, but I'm not sure I'd say it has a strong story line...

In my mind, Metal Gear Solid 4 is the best single-player game on the system. And it has a multiplayer mode.

Demon's Souls

Tekken 6. Ok so the storyline is sort of lacking, and its only been released in Japan, but the trailers and videos are AMAZING. I highly recommend it. (Oh and it comes out October 2009, that's this month).

I'd reccomend Fallout 3 if you want an engorssing, story driven game that'll keep you amused for a long time.

GTA IV is a great game to hop in and out off. Its something you can have a quick blast on or play all night if you want to.

I'm currently really enjoying Batamn Arkham Asylum, so if you like Action/Puzzle games, this is a good one to get.

Call of Duty: Modern Warfare 2 coming out 11/10/09.

According to several previews the game Uncharted 2, among thieves is going to be great.

Borderlands, hands down.