If you're looking for cartoony action adventures, Daxter is a must. The new Jak & Daxter, Ratchet & Clank: Size Matters and Secret Agent Clank are also decent. If you don't have a PS3, LittleBigPlanet might also be worth it.
You can't really call these puzzlers, but LocoRoco 1 and 2 are awesome fun, Patapon is an acquired taste in a similar art style. Exit is a pretty cool action puzzler, loads of content and quite challenging. Both Lumines games and Echochrome are also worth it.
There are some pretty decent racing games on there but I can't recommend many as I don't know much about them. Burnout Legends is probably a safe bet.
Action adventures: God of War: Chains of Olympus is a must, pushes the hardware to the limit and some really fun gameplay (if you enjoy God of War, that is). The GTA's are pretty good, although very much limited in terms of mission length/gameplay possibilities (and graphics, naturally) compared to their console counterparts. MediEvil Ressurection is pretty dated by now but still fun. Also Metal Gear Acid 1 and 2 if you're looking for something completely out there, or Portable Ops for some more conventional MGS stealth gameplay/story.
Ace Combat X for flight arcade stuff, Tekken: Dark Ressurection for beat'em ups.
In terms of shooters, check out Killzone: Liberation and Resistance: Retribution. The Siphon Filter's pretty good too.
If for some reason you enjoy JRPGs, there are some good remakes or ports out of golden oldies, like the two first Star Oceans and Valkyrie Profile, which is fantastic. Crisis Core is the only good FF7 spinoff, although mind that it's very anime-inspired. There's also remake out of Final Fantasy Tactics (originally for PSX), which is great.
Also use it as an emulator as others have suggested.
But yeah, PSP's totally useless if you're looking for CRPGs, RTS and FPS. Big surprise there! (it's not the greatest handheld out there, obviously, but I'd argue it DOES have some stuff worth play unlike what others have suggested)
Oh, just noticed:
One cool thing is apparently I can download PSone classics directly to the PSP. They have Legacy of Kain: Blood Omen and Soul Reaver, and I hear those praised a lot. Worth it for six bucks a pop?
If you haven't played them, very much YES. They're wonderful, wonderful games. Blood Omen might feel slightly dated but Soul Reaver holds up very well today. Some of the greatest action adventures I've played: intricate storyline, fantastic universe and atmosphere, moody music, huge open world with tons of optional stuff to do, and Soul Reaver has some brilliant environmental puzzles (although you'll have to put up with the sometimes annoying box sliding puzzles in return).
