Man I love the PS2. It was such an iconic system and the games just kept coming. It was an era where you could walk into a game store, close your eyes and snatch a random PS2 game and it was probably an affordably priced banger of a game.
Some of my favorite that come to mind are the two Gungrave games. Just insane shoot em up action. And they're by the same artist that did Trigun.
The Maximo series! Basically spiritual successors to Ghouls N Ghousts. Really challenging 3D platforming.
Katamari Damacy is always a super fun time, of course.
There's an Astro Boy game made by Sonic Team that is kinda mediocre but I find it so charming I don't care.
And of course Kingdom Hearts. I think Kingdom Hearts 2 wins the "Most PS2 Game" award. It's a defining game for the system and showcases everything it can do to its fullest.