No particular order (except the first one is number one)

-Metal Gear Solid 3: Snake Eater
-Tony Hawks Pro Skater Underground 2
-God of War 2
-Ratchet and Clank (all of them)
-GTA Vice City
-Prince of Persia: Sands of Time
-Kingdom Hearts 2
-Need for Speed Underground

In no particular order:

- Final Fantasy XII
- Gran Turismo 4
- Metal Gear Solid 2: Substance
- Metal Gear Solid 3: Subsistence
- Gradius V
- Dragon Quest VIII: Journey of the Cursed King

The list is smaller than I expected. But that gen was an unusual one that had my main console go from the Dreamcast to the PS2, and then the PS2 getting slightly edged out as my main console by the GameCube (like 40-60 or 45/55 %). The last time a Nintendo made a console that managed to be my main console of choice. The only other having been the SNES, two generations prior.

So there are quite a few titles that could have been included here, but I played them on either Dreamcast (like Resident Evil Code: Veronica and Street Fighter III: Third Strike) or GameCube (like Resident Evil 4 and Beyond Good & Evil).

PS2 is king

SSX3

FFX

Okami

NFS Hot Pursuit 2

GoW

GoW2

Deus Ex

Mafia

Hitman Blood Money

Resi4

Prince of Persia - Sands of Time

ATV Offroad Fury 4

WRC Rally Evolved

Haven't played PS2 games or the console as much as I should've. Maybe I'll buy a PS2 one day. Of the ones I've played my favorite are...
-Lego Star Wars: The Video Game
-Both Star Wars Battlefront games

I knew you would say that lol. But jokes aside, I really like the SSX games too. SSX Tricky and SSX 3 were amazing, I also played SSX On Tour for a short while.

On topic: So it appears that all games from my list in the OP are resonating with you guys since all were mentioned a few times in your lists. However, nobody likes FFVII: Dirge of Cerberus *cries in the corner* I wonder why.

Too many

ICO/SotC
Okami
MGS games
FF games
Burnout games
Colin Mc Rae games
WRC games
Need for Speed games
Gran Turismo games
Wipeout Fusion
Jack and Dexter games
Racthet and Clank games
God of War games
GTA games
SSX games
Silent Hill 2
Bladur's gate Dark Alliance
Champions of Norrath
Herdy Gerdy
Prince of Persia Sands of Time
We love Katamari

It's hard to believe PS2 had so many great games, did that gen last 2 decades??

Silent Hill 3
Primal
Shadow Hearts 1, 2, and 3
Kingdom Hearts 2
Tekken 5
Beyond Good & Evil
Castlevania: Curse of Darkness
Arc The Lad: Twilight of the Spirits
Radiata Stories
Steambot Chronicles
Grand Theft Auto: San Andreas

Metal Gear Solid 2/3, Persona 3 and 4, Yakuza 1-2, Dragon Quest 8, Tekken 4 and 5, Clock Tower 3, Haunting Ground, True Crime New York City, all GTA-Games, Dragon Ball Z Budokai 2/3, Resident Evil 4 and many more :D

I tried playing Dirge of Cerberus, but I couldn't get past the first boss.  The controls on that part did not feel intuitive at all, not to mention that it is nothing like the gameplay up to that point.

Shadow of the Colossus is just perfection.