Killzone Liberations is a top down strategy shooter that's actually pretty fun (at least 10x better than the generic FPS game on the ps2). You actually have to use duck and cover tactics as the AI is pretty vicious.
Syphon Filter: Dark Mirror remains as the PSPs "unheard of gem." It's seriously one of the better 3rd person shooters I've played on ANY system in a long while... then Gears of War came out so it's the best 3rd person shooter on any handheld console.
If you like turn-based strategy, get Field Commander. The loading times are a little slow and it's a plays a lot like Advance Wars but it's incredibly in-depth and has a plethora of multiplayer modes plus a mission/level editor.
I also like Generations of Chaos: Aedis Eclipse which is miles ahead of the first GoC. Great turn based strategy there.
Ratchet and Clank: Size Matters is a great addition to the series and plays just like the console versions. Death Jr 2 is another excellent platformer/shooter.
I have about 30 PSP games but those just happened to cross my mind. There's a couple of good looking ones coming out this year like Dungeon Maker (you build a dungeon to capture monsters and then send adventurers in to hunt them down), Dungeons and Dragons Tactics (which uses the 3.5e ruleset), Brave Story (a pretty decent looking RPG), Disgaea Portable (which has a plethora of extra features), Final Fantasy Tactics S (NEW TRANSLATION GOD YES!), Joan D'Arc, AND Castlevania: Rondo of Blood (a remake of Dracula X as well as A FULL VERSION OF SYMPHONY OF THE NIGHT... 2 games for 40$ YES PLZ)
Seriously, I just want Disgaea Portable. It has ETNA MODE, a what-if scenario where Etna kills prince Laharl in the beginning and tries to take over the demon kingdom. I also can't wait for FF Tactics; Tactics is my favorite Final Fantasy game but the translation was downright GARBAGE so a new tran will be good.
As far as Homebrew goes, it's only sweet, sweet icing on the cake. Don't let anyone tell you it's garbage. No other handheld system lets you play all sorts of roms, SCUMMVM (yes, I play MONKEY ISLAND on my PSP with perfect speed), as well as PS1 games. There's also several accessories available like a wireless tv tuner which sends RF signals so you can watch TV on the PSP with no cables, a GPS tracker (why you'd be lost in the woods with a PSP is beyond me), a DIVX encoder (there's actually several dozen homebrew encoders available like AVI and mpeg1-4), and even adaptors that let you play your PSP on a large screen monitor.
Seriously, PSP is the best handheld available in my opinion. The only thing that sets it apart from the DS is that Nintendo has killer 1st party support with a lot of strong titles. Simply having "Zelda" or "Pokemon" in the title is INSTANT MILLION DOLLARS.