The games I personally enjoyed most on PS2:
Disgaea: Hour of Darkness.
Ace Combat 4: Shattered Skies. I assume the reason I didn't see this in every post is some kind of malfunction in my laptop.
Katamari Damacy. Pure joy in videogame form.
Dark Cloud 2. THE underappreciated gem of the PS2 generation.
Prince of Persia: The Sands of Time. So much more than environmental puzzles and combat. Beautiful but also somber, with a surprising story and two main characters you actually want to hear talk to each other.
Resident Evil 4. The best Resident Evil game of all time, and I say that as someone who really loves resource management. Perfect pacing, full of variety. Everything is so polished and there are so many tiny extra touches that it starts to feel a little Kojima-esque.
DMC3:SE. This game. I still remember getting it home, and watching the trailer that plays on the menu screen, where it shows clips of each different ability. Wait, you can surf on top of enemies? You can STOP TIME?
Incredibly deep, tons of fun, creative bosses, top-notch graphics, attitude to burn. The mission structure and scoring system give it replayability GOW can only dream of. My favorite game on PS2.