PSP has a substantial library, that will satisfy any gamer, but most games, that get the spotlight, well, they suck. Here is my pick of good stuff:

Street Fighter Alpha 3
Blazblue II (same as the 1st one, but with more characters)
Dissidia 012 (the best game I played in years)
Darkstalkers
Final Fantasy (Crisis Core is awesome)
Kingdom Hearts
Star Ocean
Popolocrois (rare, but very good)
Midnight Club
Metal Gear Acid (make sure you check both)
Castlevania (the best!)
Pangya
Jeanne D'Arc
Phantasy Star
Y's Seven
Fat Princess
Tactics Ogre
3rd Birthday (loved the concept)

If you like JRPGs, then it's one of the best systems to grind 'em, you have a lot of Final Fantasy games, Gungnir, Breath of Fire, Y's, Generation of Chaos, Trail in the Sky and many more. I'm talking months of pleasant gaming experience. I fly a lot because of my job and the only panacea for jet lags is grind!

Quote from Buried On Mars on January 4, 2013, 09:33
Nice list wildbird! I liked 3rd Birthday. Got a little to grindy towards the end to beat the end boss, but good shooter/PRG.

Yeah, the game is great, but too many unlockables, trophies and shit. This game could have been made into a great movie.

Psp is for sure under-rated in my book.
If your more into puzzles, i'd check out both Exit, and Mercury... very unique puzzle games. Also, for more classic on the go, there was a namco museum, and midway arcade treasures released for the psp as well as the sega collection (a slimed down version of Ultimate Genesis Collection)
the Grand Theft Auto games also play very well, and even look a bit better than on the ps2, plus if a friend has a psp and the same gta, you can do multi-player.
Lastly, Street Fighter Alpha 3, Wipeout Pulse, and Grand Turismo play perfectly on the system. N+ is also great for a challenging arcade platformer.