Aqua Teen hunger Force: Zombie Ninja Pro Am, Everblue 2, Gran Turismo 3, Jackass, Metal Saga, Rumble Racing, Tokyo Xtreme Racer 3, and Tokyo Xtreme Racer Zero all come to mind when it comes to PS2 exclusives that are a lot of fun to play.
Loved that one as well. For some reason I didn't have any friends that got into it.
I mostly enjoyed looking for the Easter eggs to unlock new vehicles, but the races are pretty intense at the Elite Cup levels. For those that haven't played it, I'd recommend picking up a CIB because it's a fun, cheap, arcade racing game that also supports two-player.
One of my favorite games on PS2 that seems to get overlooked is Primal. It was made by Sony but was kinda unique at the time. I like the story and the main character reminds me of my ex. The combat is kinda shallow and repetitive but the atmosphere is great.
PS2 will be the next big thing to collect for. The games have gotten more expensive than they used to be. Soft modded PS2 fat with a hardrive or a modded ps4 is the way to go.