We need an X360 thread, too. :)
I'm personally really hooked on this system. And after replacing my launch model with a Jasper (newest revision), I don't have to worry about RROD anymore, and a 120GB harddisk reduces disc drive noise to almost nothing.
With the old Xbox hardly having any interesting games, I've been really pleased with 360, which lifted the heritage from Super Nintendo and PS1 and 2, with a wide range of varied games, a lot of exclusives and some of the most interesting Japanese releases.
Some of my favourite games:
Bioshock
Lost Odyssey (very underrated IMO - it's classic Final Fantasy when it's best)
Dead Rising
Oblivion
Gears of War 1 & 2
Prince of Persia
Street Fighter IV
Ghostbusters: The Video Game
Assassin's Creed
Fable II
And the following are highly recommended on Live Arcade, you should get them all:
Castle Crashers (great Golden Axe clone, but way better than the games that inspired it)
Shadow Complex (great Metroid clone)
Prince of Persia remake (nice improvement with full respect to the 1989 classic )
Castlevania: Symphony of the Night (identical to the PS1 version, but if you don't have it, get it, my favourite game of all time)
Braid (everyone else probably already told you to get this, it's a bit short for the price, but everyone should try it, it's like no other game ever made, and makes you use your brain in ways you never used it before)
Rez HD (HD remake of the Dreamcast/PS2 classic, a GREAT music and art game that probably goes well with psychedelic drugs. If you don't have the original, get this version. If you loved the original, get this as well)
Trials (pretty unique motocross/stunt game running on a strictly 2D axis. Great for comparing scores with your friends - a LOT better than it sounds, try the demo)
Monkey Island SE (THE adventure game classic :))
Bomberman Live (it's Bomberman, on Live!)
Puzzle Quest 1 and 2 (who ever thought combining Bejeweled with RPG type battles and stat building could be so much fun? EXCELLENT game)
Mega Man 9 (only for the hardcore! Classic old school fun)
Bionic Commando Rearmed (another excellent remake, best example of a remade 8 bit game I can think of. A classic)
Pac-Man: Championship Edition (I never enjoyed Pac-Man much, but this version make the classic gameplay action paced and tons of fun)
Ikaruga (another straight port, but one of the best 2D shoot'em ups I've played. Very unique)
Post your Live IDs if you care. Mine is, obviously, "Sumez".
http://live.xbox.com/member/Sumez