Fire Team Bravo 2, that campaign lasted me a while and I liked getting points for my multiplayer stuff while playing single player
Only the multiplayer was fun. Also, probably one of the most fun multiplayer games on the PSP. I'm debating whether or not I should buy a second PSP (I modded mine to death) to get the FTB3. Looks fun.
Also, get
Monster Hunter Freedom: Unite! That is actually the trailer from the first game (All ingame footage), but each sequel just builds on the previous game. It has been, by far, my favorite PSP game of all time. So much fun, so much to do, and such great times. And it's hard. Also, the majority of the game consists of "boss fights". Anyone that's ever played it will suggest it to you in a heart beat.
As for other fun games:
Any of the "Starwars Battlefront" games.
Me & My Katamari
SOCOM: Fire Team Bravo 2/3
If you can get import games, any of the Gundam games are very fun. Unfortunately it's only in Japanese.
Any of the GTA games.
I know there are a ton more, but none come to mind at the moment. It's been forever since I've last played my PSP.