I assume this thread is meant to be called "CAN'T live without" so I'll fix that for ya.
Anyway.
Lost Odyssey is an absolute must-have if you're a JRPG fan, and on the same note if, you liked Fable or love Western RPGs, get Fable 2. Oblivion also runs very well on the Xbox, as does Fallout3.
Beautiful Katamari is a bunch of mindless fun, but doesn't have quite the sparkle as its predecessors did.
Left4Dead is for any online shooter fan that loves a good basic co-op experience. Even with the "Same levels over and over" each playthrough is different, and teaming up with random people can be loads of fun.