Here are my PSP recommendations. I've been using this list for a while
RPGs
Brave Story: New Traveler
Breath of Fire III (Euro Import)
Disgaea: Afternoon of Darkness
Final Fantasy (If you like the original one and don’t already own it, best version)
Final Fantasy II (If you liked II before and don’t already own it, best version)
Final Fantasy Tactics: The War of Lions
Final Fantasy VII: Crisis Core
Gurumin: A Monstrous Adventure
Jeanne D’ Arc
Monster Hunter Freedom
Monster Hunter Freedom 2
Monster Kingdom: Jewel Summoner (I personally love it, but not for everyone)
PoPoLoCrois (I personally love it, but not for everyone)
Princess Crown (Asia Import)
Riviera: The Promised Land
Tales of Eternia (Euro Import)
Tales of the World: Radiant Mythology (The dungeon exploring may turn some off, but combat is fun)
Valkyrie Profile: Lenneth
Wild Arms XF (some will like it while some will not)
X-Men Legends II: Rise of Apocalypse
Fighters
Darkstalkers Chronicle
Guilty Gear Judgement
Naruto: Ultimate Ninja Heroes (If you are into Naruto)
Powerstone Collection
Street Fighter Alpha 3 Max
Tekken: Dark Resurrection
Puzzle
Bomberman
Crush
Every Extend Extra (short, but fun)
Exit
Gunpey (This game is tough, but interesting concept IMO)
Lumines
Lumines II
Me and My Katamari (controls are weird at first, but still good IMO)
Mercury Meltdown
PQ Practical Intelligence Quotient (It can be tough, but cool concept)
PQ2: Practical Intelligence Quotient 2
Puzzle Quest: Challenge Of The Warlords
Racers
ATV Offroad Fury Pro
Burnout Dominator
Burnout: Legends
Crazy Taxi: Fare Wars (Those who like the first Crazy Taxi may love this, but worth $20 or less)
Outrun 2 Coast to Coast
Pursuit Force
Pursuit Force: Extreme Justice
Race Driver 2006
Ridge Racer (or import 2)
Sega Rally Revo
Wipeout Pulse
Wipeout Pure
Action
Castlevania: Dracula X Chronicles
Dead Head Fred
God of War: Chains of Olympus
GTA: Liberty Stories
GTA: Vice City Stories
Killzone: Liberation
Mega Man: Maverick Hunter X
Mega Man Powered Up
Metal Gear Solid: Portable Ops
Metal Gear Solid: Portable Ops Plus
Medal of Honor Heroes (some like it)
Prince of Persia Rival Swords
Socom: Fire Team Bravo
Socom: Fire Team Bravo 2
Syphon Filter: Dark Mirror
Syphon Filter: Logan’s Shadow
The Warriors
Twisted Metal Head On
Ultimate Ghosts ‘n Goblins
Survival Horror
Silent Hill: Origins
Flight Shooter
Ace Combat X: Skies of Deception
After Burner: Black Falcon
Platformer
Daxter
Death Jr 2: Root of Evil (not for everyone, but definitely better than the previous one)
LocoRoco
Ratchet & Clank: Size Matters
Secret Agent Clank (some will like it, some may not. Play demo first)
Tokobot (not for everyone, but fun IMO)
Sports
Hot Shots Golf Open Tee
Hot Shots Golf: Open Tee 2
MLB 07: The Show
MLB 08: The Show
Virtua Tennis World Tour
Virtua Tennis 3
Winning Eleven: Pro Evolution Soccer 2007
Music
DJ Max Portable (import)
DJ Max Portable 2
Gitaroo Man Lives!
PaRappa The Rapper (The game may be too old school for some, but if you liked the series in the past, you may like this)
Patapon
Strategy
Field Commander
Lemmings
Metal Gear Acid
Metal Gear Acid 2
Sid Meier’s Pirates!
Classical Compilation
Activision Hits Remixed
Capcom Classics Collection Remixed
Capcom Classics Collection Reloaded
Capcom Puzzle World (Super Puzzle Fighter II Turbo makes it worth it)
Gradius Collection
Metal Slug Anthology (load times are terrible, and some slowdown, but the collection is excellent)
Namco Museum Battle Collection (IMO, best Namco collection)
Parodius Portable (import)
Salamander Portable (import)
Sega Genesis Collection
Taito Legends Power UP
Twinbee Collection (import)
Download (I still need to play more):
Beats
Castlevania: Symphony of the Night (If you choose to not by Castlevania Chronicles) (PS1)
Crash Bandicoot 2: Cortex Strikes Back (PS1)
Echochrome
R-Types (PS1)
Syphon Filter: Combat Ops
Wild Arms (PS1)
In the future:
Final Fantasy Dissidia
Kingdom Hearts: Birth by Sleep
Naruto: Ultimate Ninja Heroes 2
N+
Star Ocean: The First Departure
Star Ocean: Second Evolution
Ygddra Union
I'll update the list some time with more recent imports and looking through other newer releases I might have missed.