[Cue 5-page debate on "console exclusives"]
In all serious, though: Do you have a gaming PC and/or prefer to run games on a PC? Assuming that you do, then the best 360 exclusives that don't also show up on the PC are:
Dead Rising: Zombies + GTA + a whole bunch of weapons. Not my style of game, but a lot of people seem to love it.
Halo 3: Overhyped, yes, but still a fun little FPS.
Rez HD: The Dreamcast classic, in 1080p. The only "must-download" from XBLA that I can think of.
Kameo: This game gets trashed a lot on the Internet, but don't let that dissuade you. It's a beautiful 3D Zelda-like, and its only major flaw is that it's almost criminally short.
Earth Defense Force 2017: You're a soldier. You choose two out of hundreds of weapons (all of which have infinite ammo) and wade into battle against armies of giant insects, flying saucers, and War-of-the-Worlds-style tripods. Good, mindless fun - and it has local co-op.
Assuming you'd prefer to play your games on a console rather than PC, I'd also suggest:
Viva Pinata: Sort of like Animal Crossing + Pokemon-style breeding + pinatas. More fun than it sounds, or so I hear.
Condemned Criminal Origins: A scary-as-hell first-person survival horror shindig.
F.E.A.R.: Not really an exclusive, but it might as well be considering how horrible the PS3 port is. A creepy, stylish FPS.
"'Casual games' are something the 'Game Industry' invented to explain away the Wii success instead of actually listening or looking at what Nintendo did. There is no 'casual strategy' from Nintendo. 'Accessible strategy', yes, but ‘casual gamers’ is just the 'Game Industry''s polite way of saying what they feel: 'retarded gamers'."
-Sean Malstrom