Ratchet & Clank series were my favorite games on PS2. Definitely check them out.

Burnout games.
Colin McRae games.
SSX games.
Wipeout games.
Baldur's gate dark alliance and Champions of Norrath.
Katamari Damacy
Herdy Gerdy
Final fantasy 12

And of course all the stuff already mentioned and remastered like Okami, Silent Hill 2 (better on ps2), R&C, J&D, GTA, MGS, GoW, Ico/SotC

You reccomend the first R&C, but not the vastly superior 2&3??

The Tenchu series (Wrath of Heaven and Fatal Shadows are both on PS2) is a really good ninja stealth game.

Really surprised nobody mentioned Dragon Quest VIII !!!

You beat me to it!

I have the best taste in games and my fav PS2 games are SSX3, FFX, Okami, Deus Ex, Mafia, God of War 1&2, Prince of Persia The Sands of Time, Need for Speed Hot Pursuit 2, the Splinter Cell games, the Hitman games, although some of them are better played on other platforms.

I put the one I remember the best but the whole trilogy is recomendable.

Final Fantasy X, Kingdom Hearts 1 and 2, Ratchet and Clank series, Jak and Daxter series.

Most obvious answers have been said already so I will add:

Dynasty Warriors 3-5
Romance of the Three Kingdoms 7-11
Dynasty Tactics 1-2
Samurai Warriors 1-2
Onimusha
GTA 3, VC, SA
Bully
Odin Sphere
Twisted Metal Black
Monster Rancher 4