I know this post is about a PS2, but looking at that old, enduring console of yours made me miss my PS1.

Anyway, some of my personal preferences you might like are...
GTA: San Andreas is a must. Add Bully if you like the trait of that game. Shadow of the Colossus and ICO for something 'deep'. I heard the former has a terrible remake on the PS4, so enjoying the PS2 version is better. And also Guitar Hero 3: Legends of Rock, but only if you're into rhythm game. 👍