You got a PS2? The world is ending. XD; Well, my recommendation list:
Tales of the Abyss
→ Great story, characters, gameplay, everything. Get it if you can find it)
Final Fantasy X
→ If you're into turn-based RPGs, this is a good choice. It has a veeeery clichéd plot (what JRPG doesn't? XD) but not bad. Probably a must-have on the PS2 anyway. :(
Final Fantasy XII
→ Awful story (or none at all, take your pick), boring characters, but I'm still recommending it because the gameplay is pretty fun. :D
Katamari Damacy / We ♥ Katamari
→ Get it get it get it get it get it. It's weird, but it's so SO awesome.
Okami
→ You probably already have it for Wii, but if you don't, try to find the PS2 version instead. It's identical but without sloppy Wii remote controls. XD
You've already listed Kingdom Hearts so those would be my must-haves. I am an RPG fan so the list is rather biased but... the games in other genres I do own... haven't been played so I can't recommend them. :P
Other games that are generally regarded highly:
→ God of War
→ Persona 3
→ Dark Cloud 2 (I love the first one but most people hated it--this is apparently much better)
→ Shadow of the Colossus
→ Psychonauts
→ Disgaea
→ Ico
Look 'em up and see if they pique your interest. I own a fair few of these but with the exception of God of War and SotC, I haven't touched them yet. XD I just plain sucked at both of those but they are good games, from what I played and what everyone says.