My must-have reccomendations:
Sports:
Pro Evolution Soccer 4 (amazing game, blows FIFA out of the water)
Amped 2 (harder but more rewarding and realistic than SSX)
Top Spin (best fun you'll have with friends besides PES4)
for "American sports" most ESPN games are awesome
Action:
Ninja Gaiden (HARD but pretty)
Otogi (can be had cheap and is really good)
Panzer Dragoon Orta (style in perfection, little odd though
)
FPS:
Halo 1 & 2 (need I even comment?)
I prefer FPS on PC, so no others here
RPG:
Fable (ignore the hype and bashing, just enjoy it!)
Morrowind GOTY (re-discovered this game on Xbox recently, I find it more enjoyable than on PC!)
SW: KOTOR (best SW game ever)
Beat em up:
DOA3 (not deep but pretty and FUN!)
DOA Ultimate (much more of the same)
Soul Calibur 2 (more depth but I find the character design a bit silly)
Racing:
Rallysport 2 (arcade racing at its best)
PGR 2 (reference on Xbox)
Burnout 3 (crash and burn)
Other:
Beyond Good & Evil (surprisingly good Action Adventure, loved it!)
GTA Double Pack (old games, but still cool to pick up and play every now and then!)
I probably forgot some games, but those should keep you busy for a year or two!