I have still not played Trails in the Sky. Given my love of Falcom games I really need to get on this bandwagon. Maybe after I finish replaying Xenogears again and Y's Celceta.
There was a Carnage Heart game for the PSP!?!?!?!? I really, really, really need to find a copy of that. Had no idea at all!
My personal recommendations would be:
Ridge Racer 2 - I think you have to import this one. It's basically a re-hash of the first PSP Ridge Racer (same engine, same intro movie, same menus), but instead includes the tracks from pretty much every good RR game - All tracks from Ridge Racer, Ridge Racer Revolution, Rage Racer, Rave Raver, and R4 are here. As Ridge Racer games go, it really is the best one - only flaw is that it lacks R4's yellow-ass menus and style.
Ace Combat: Joint Assault - Great portable Ace Combat game with co-op multiplayer (even online if the server still exists).
Phantasy Star Portable 2 - Like a portable version of Phantasy Star Universe. Loads of content, lightning fast load times if you install to memory stick, and online play (again, if the server's still there).
Gods Eater Burst - Like a story-driven Monster Hunter but a bit more forgiving gameplay-wise, and with the inclusion of AI teammates, which is a boon when you don't have friends around.
There are other games that I'd highly recommend, but most of them have already been mentioned in the thread.