I have a few sentimental favorites.
Spiro: enter the dragon. First game I think I ever put over hundred hours into. I beat every challenge and mini game. Found every secret, and even remember telling a guy at block buster how to get past one of the final levels. Just the mere mention of block buster should give you an idea of how long ago this was XD. The game even broke on me, it was thinking it was a pirated copy. I took this as a challenge and beat it again. I remember hitting the final boss, getting teleported back to the final stage, and calling it a win.
Star wars: battle front two . Admittedly, I played it more with my psp then with ps2, but ps2 is where I found it. I originally played with a friend of mine, doing some couch co op. We played it on a fifty inch screen, because for some reason, the sound would work for video games, but not for tv channels. I remember playing on hoth as one of those giant yeti guys (can't remember their name), and laughing while taking down some storm troopers with nothing but claws. Even being on endore and taking one of those storm trooper walkers.
Digimon rumble arena two . Another game I did a lot of couch co op with. A friend of mine and I had rented it so many times, I don't know why I didn't buy a copy. Our pvp matches were always amazing. Typically coming down to the last five to ten percent of health bar. I always loved using that fox digimons (another name I can't remember >.>) last form and final move to snare my friend when he tried to attack me. I am so getting a copy when I have some extra cash.