I got a PSP for my 21st (and a PS3 - but no problems getting good games on that front!) with LocoRoco (which I've almost completed - fantastic game!) and Patapon.
I'd quite like to get more games for it (having only 2 games for a system just feels ...wrong...) but I'm not sure what's out there and any good?
(I like all genres, though RPGs/Beat 'em Ups/Puzzlers/Rhythym-something (i.e. Lumines, Rez etc.) kinda games are my favourites if pushed to choose.)
I play the crap out of Hot Shots Open Tee. I just picked up Monster Hunter Freedom 2, but haven't played it yet. It is kind of in the vein of a 3D Diablo/Phantasy Star Online game.
I just purchased a PS over the weekend...I picked up the Gof of War edition...man GOW is awesome on the PSP...two thumbs up...I also grabbed Crisis Core and Puzzle Quest...havent played much of those but CC seems pretty awesome as well
There are really a lot of good games out there for the PSP. Some of them are older titles, which I'll list here (since most of the newer ones have already been mentioned).
Ones I like: Sony's Untold Legends (both of them, far more Diablo-like than the aforementioned Monster Hunter Freedom 2, which is just... odd), and Sid Meier's Pirates! A classic, even if it is on the little screen.
You didn't mention tactical games as part of your usual fare, but I do have three suggestions: Final Fantasy Tactics: War of the Lions (a great game), Dungeons & Dragons Tactics, and Field Commander (for the turn-based enthusiasts).