There are a lot of enjoyable 360 exclusives. I would list them for you but you see...I wasn't born yesterday. :O --- Welcome to Gamefaqs, where for some reason it's ok to bash Americans continually without fear of penalty. I guess we don't count as people here.
Condemned - Probably the only legitimate horror game I've played this gen. It's well worth a play while waiting for Silent Hill Downpour.
Dead Rising - My favorite actual exclusive on the system. It has great presentation and humor, and it just felt very unique overall.
Lost Odyssey - It sort of felt like a Japanese take on Planescape: Torment... which is a good thing.
Shadow Complex - Metroid Gear Solid... need I say more?
Crackdown - Not as good as Infamous or Prototype... but it's a sandbox superhero game. By definition, it's good! If you're like me, you'll spend hours just upgrading your jump abilities and leaping over skyscrapers. It can be a lot of fun.
Limbo - Great puzzle-based platformer though it's quite short... and I'm pretty sure it's getting getting released on the PS3.
Tales of Vesperia - One of the reasons I bought a 360. If you've played one Tales game, you've played them all... This is one of the better ones though.
Mass Effect - The other reason I bought a 360.
Super Meat Boy - One of the best (and most frustrating) platformers this gen. I'd probably put it in my top three.
Divinity 2: Dragon Knight Saga - I really enjoyed this game... probably just as much as Infamous 2, and that was my most anticipated game of 2011. The mind reading gimmick is a lot of fun and the writing in general is great.