kirbyTomb Raider (pretty much all of them for PS1 and PS2)
SSBM
Goldeneye 007 (1997 but fuck you)
Pokemon Red all the way to Black/White (haven't played B/W2)
Ratchet & Clank
Resident Evil 2 (don't remember if it was made in the 00's) + Resident Evil 4
Final Fantasy X/Final Fantasy X-2/Final Fantasy XII
Star Ocean 3
Grand Theft Auto 3/Grand Theft Auto VC
Spyro
Ocarina of Time/Majora's Mask
Kingdom Hearts 1 + 2
TF2
Age of Mythology (don't remember much of this game but I remember enjoying it)
Portal 1
Skyrim
THPS (1999 but whatever)/THPS 2-3/THUG 1 + 2
Call of Duty 4/Call of Duty 5
Red Faction 1
Darksiders 1 + 2
Dishonored (really amazing game, best singleplayer I've played in awhile aside from BL2)
Borderlands 1 + 2
Rayman 2: The Great Escape
Mario Party 1 + 2
Shadow of the Colossus
Gran Turismo 4
NFS: U 1 + 2/NFS: Most Wanted
Burnout 1 + 2 + Revenge
TimeSplitters 2 + 3
I'm pretty sure I'm forgetting a lot more, but these have all been memorable/amazing games that I've loved.
TimeSplitters was my all time favorite game of the 00's hands down.