Yay! I'm glad you're asking for recommendations on the PS2; that's the only console I can really recommend any games for. Here we go:
-Okami. This'll take you ages to complete, not counting all the things you can collect
-Grand Theft Auto: San Andreas/Vice City. Either of these games are really fun.
-Gran Turismo 3 A-Spec: a very good racing game.
-Crimson Tears: engrossing storyline and great battle system.
-God of War: definitely get this if you haven't already. Epic!
-Rez: hard to explain how this game is, I suggest looking on youtube~
-Outrun 2006 Coast-2-coast: Racing game, I think you'll enjoy it
-Jak and Daxter The Precursor Legacy: the first game in the series of J&D games. If you don't like this one, you might not like the others, but the other games in this series feature guns and flying vehicles, so...if you like that, go for Jak II!
-Ratchet and Clank: great weapons, storyline, scenery and humorous dialogue.
-ICO: a lonely boy and girl have to escape a prison via puzzles. Very enticing.
-Tekken 5: pure fighting, very fun, doesn't really need an introduction
-Dragon Quest Journey of the Cursed King: very fun game and will take you ages to complete.
Hope this helped you a bit!