Looking at my library of Xbox 360 games that I have played and still own....
Any of the Dead Space games would be great choices. Dead Space 1 in particular is a classic survival/horror game that always feels like a cross between Alien and John Carpenter's The Thing to me.
Dishonored is another strong single player adventure where your choices really affect the game world and ending.
Batman: Arkham Asylum and Batman: Arkham City are probably two of the strongest comic-book inspired games out there.
I actually preferred Bioshock 2 to the original as it had a better story, the underwater setting factored into the gameplay, and the ending was more satisfying (the original Bioshock has one of the worst endings of this generation).
Enslaved: Odyssey to the West is another very good action/platforming hybrid with a very well presented story and some of the best voice acting in a game this generation.
Fallout 3 is a great RPG with strong action elements and if you enjoyed Oblivion's style of gameplay but want to experience something similar in a post-apocalyptic environment, this may be a good choice.
Any of the Forza Motorsport or Project Gotham Racing games would be good options if you're looking for a racing game - lots of cars (and motorcylces in PGR4) and varied tracks and locations.
Nearly all of the above games can be had for under $20 each new and that that price I'd say they're just about all keepers.
Edited by TonyDP - 2/18/13 at 12:28pm