I don't think I fit the kind of appeal the PSP targets, so I've never found many games I care for myself... however someone brought up Sonic Rivals, and I've played the second one. I would like to note that you aren't going to get much out of the game if none of your friends have the game, because singleplayer gets tedious after a while especially because (aside from that stupidly overpowered ESP move and the stupidly underpowered tails flight) there is no difference between characters and the dialogue is seldom worth playing for.
The fact that it does not have infrastructure makes it even more problematic, because it seems to me like it would be a terrific game to play with a friend, but unless your friend has a PSP, you can't play the battle stages, and unless your friend has their own copy of the game you can't play races.
I can heartily recommend Ratchet and Clank: Size Matters as quite a decent hack-n-slash/platformer hybrid, but I think I've heard that it is also on the PS3. I've heard some great things about Daxter, but haven't tried it... I can also say that I've not had a very good experience with FPS games due to the small camera. Battlefront is only good for singleplayer (imo) because its lock-on system is abused to no end in multiplayer, making it almost harder to get into than Prime: Hunters (and definitely feels more cheap).
