I thought the Sly Cooper games were totally badass. A world in which I am a master thief, breaking hearts and locks all over the planet? Yes please! The games were clever, funny, epic, and followed an entire story from start to finish without any "to be continued" bullshit(except at the end of every in-game episode, haha!).
I never thought I'd say it, because for some reason I thought they were stupid before I played them, but the Ratchet and Clank series. I wouldn't play them again, due to how much unlocking there was to be done, but what a ride. I was damn good at it, too. I guess I just didn't like that the main character seemed to be a furry. See comments above regarding Sly Cooper.
The .hack series was a whole chapter of my life unto itself. After seeing .hack//SIGN, I immediately ordered the .hack series, but the last game was rather grind-tastic so I just about lost interest. Then .hack//Roots and the accompanying G.U. games came out and I did the whole dance again, but without the dropoff. Goddamn fantastic games, 6 of them!
I never played ICO, but I want to so bad it hurts. I would say that it and Shadow of the Colossus are irreplacable PS2 gems, but they're being remastered on PS3 to not have that weird fuzz effect and much better draw distance, etc. etc. so scratch that. Get'em for PS3.
Don't forget the horror! The PS2 had an as-of-yet totally unmatched horror scene. Mine doubles as my PS1 as well so it can play Silent Hill 1, but also Silent Hills 2, 3 and 4 as well as Origins (which is awful for a SH game but still pretty good). There are more B-movie horror games like a lot of RE games (including Outbreak file #1 and #2, which I actually enjoyed the hell out of), Cold Fear, Obscure and The Thing. Finishing the list off the top of my head is The Suffering, Siren, Haunting Ground and the wonderful Fatal Frame trilogy.