Although it's from back in 2005, "Need for Speed: Most Wanted" was some of the most fun I've had in a driving game, ever. Because of its age, you could probably pick it up used for pretty cheap. I posted a thread about the demo when it came out, and I played the full game all the way through. This was on the PC, of course, but the 360 version should be nearly identical.

For other genres, Assassin's Creed and Call of Duty 4 are getting high praise, so those are probably the ones I'm going to pick up next.