So much I haven't played, but from the few I have, have to say Okami.
Followed by Ratchet & Clank 2: Going Commando.

Additionally a little personal favorite: Robot Warlords (aka Velvet File).

Persona 3 and 4, MGS2 and Tales of the Abyss.

(Technically also Tales of Symphonia ofc but that will always be a GameCube game in my mind)

With no specific Order:
Metal Gear Solid 3 ... Music, Story, Graphics, Gameplay.
Shadow of the Colossus ... Music, Story, Gameplay
Resident Evil 4 ... Gameplay
Legacy of Kain Defiance ... Graphics, Gameplay, Story
Soul Reaver 2 ... Gameplay, Story
Silent Hill 2 ... Music, Story
Final Fantasy X, ... Music, Story
Final Fantasy XII ... Gameplay, Graphics, Music

Where do I even start? Final Fantasy X, Sly Cooper 2, Ratchet and Clank: Up Your Arsenal, Final Fantasy XII, GTA Vice City, Kingdom Hearts, Jak & Daxter 2, Dark Cloud, Rogue Galaxy, Okami, GTA San Andreas, so many classics and under appreciated gems!

I think my all time favorite PS2 game would be FFX. Nostalgia plays a huge part in it, but it was one of the first games I've bought and beat on PS2, and it got me hooked onto JRPG's.

Kingdom Hearts 1 or Jak 3. Nothing else comes close to the sheer greatness of those two.

Soul Calibur 2
Marvel vs Capcom 2
Devil May Cry 3
Kingdom Hearts 2

Off the top of my head.

My favorites are Final Fantasy X, Resident Evil 4, God of War, Ratchet and Clank, Grand Theft Auto Vice City, Burnout Revenge, NFS Hot Pursuit, Persona 3 and many others! PS2 is my all-time favorite console. If I had to pick one to call the best PS2 game ever, I'd probably say Final Fantasy X.

1 RE4
2 MGS3
3 GoW2

Really could have made close to a month worth of new polls with all of the games you guys listed and so many classic games in the mix too. Even some obvious ones that I should have included like Disgaea and Marvel vs Capcom 2. I picked the ones that were mentioned the most on here. Devil May Cry and Jak and Daxter just barely missed out.

On that list, probably GTA Vice City, but my real answer is Xenosaga Episode 1