These threads can be annoying, but this is actually a good topic, just because of how varied the PS2 library is. My God, I've had one for six years and my collection has spilled off the shelves and into stacks on the floor. It's the first console I owned tht I was old enough to work and buy games for the entire time.
Well, here's my list of top 10 PS2 exclusives (with some commentary):
1.) Devil May Cry 1 & 3 - The first one is sort of forgotten, but was really revolutionary at the time, at least for me. It was the first time I really saw games start to show the production values of movies. It had incredible atmosphere, sense of cool (as someone said), and a decent fighting system. The third one has the best cutscenes ever, IMO, and the customization available with the fighting...whew, I felt like the shit playing that game. DMC 2, on the other hand, was good but rushed and generic and blah...
2.) Kingdom Hearts 1 & 2 - A lot of people were surprised these games came out so good, but if you followed it from early development to final release, it looked fantastic the whole time. Played good, even if the story sort of collapses in on itself, as FF games tend to do.
3.) Bully - Best game I played in '06. The whole "sandbox" thing never intrigued me, but this took it to a different level. Great graphics, music, and a darkly funny, surprisingly mature story. Great original music score. Just criminally underrated.
4.) Katamari Damacy - Can't say enough about this one. Perfect marriage of simple gameplay made more complex by great execution. Best soundtrack ever. One of the best opening videos ever. Anything that has the King of All Cosmos vomiting rainbows (and cows) is killer.
5.) Shin Megami Tensei: Nocturne - Got this (apparently rare) game w/guidebook for $25. Best value ever. A little too weird for its own good sometimes, but the press turn battle system...so simple and so complex. Fusing demons could have been a game by itself. One of my top 3 RPGs ever (Final Fantasy VIII and Chrono Trigger are the others).
6.) Okami - People "in the know" about offbeat titles love this one, almost to the point where its overrated. But damn is it good. The PS2 Zelda as people like to say, and, at times, surpasses even that revered series.
7.) Sega Genesis Collection - Best retro comp ever. So many gems: Comix Zone, Ristar, Gain Ground. All packaged and presented perfectly.
8.) Shadow Hearts, SH: Covenant, SH: From the New World: These titles get plenty of respect from RPG fans, and rightfully so. Spent a good afternoon tracking down all three in one day. Found the first one at a flea market for $15. Second at a different booth for $10. Got the third from EB after the price drop to $19.99. All three for the price of 1 game and I would have paid $50 for each one. This from a hardcore CAG. High praise.
9.) La Pucelle Tactics - IMO, the worst game on this list. BUT its long, its deep, and the best part is its atmosphere. Besides Katamari and Okami, its the most artistic here and on the PS2 in general. Pretty humorous, too.
10.) Amplitude - A lot of people say Frequency is superior, but the song selection here is just awesome. Incredible track from Bowie still lodged in my head and its been a couple years since I picked this up. Nothing in gamedom matches the feeling of getting really locked into a song.
Honorable mention: Final Fantasy XII. Final Fantasy X2 (I am the only person who liked this, I think. Too much estrogen, but still great). Shin Megami Tensei: Digital Devil Saga 1 & 2. Suikoden III. Dark Cloud. Xenosaga. Twisted Metal Black (what the

happened to this series???). Virtua Fighter 4. SMT: Devil Summoner (not nearly as good as the other Shin Megami Tensei games - doesn't mean its not good). ICO. Makai Kingdom. Disgaea.
I could go on all night (and I already have).