Legend of Heroes: Trails in the Sky is my favorite PSP game. More people should buy it and encourage XSeed to localize the Second Chapter. I need the second half of the story!
Kingdom Hearts BBS
Corpse Party Blood Covered
Corpse Party Book of Shadows
Mahou Shoujo Lyrica Nanoha A's Portable 2: The Gears of Destiny (really good fighting game with simple controls)
Fate/EXTRA
Project Diva 2nd
Project Diva Extend
Conception: Please have my children! (needs English patch)
Queens Gate: Spiral Chaos (needs English patch)
Black Rock Shooter (is there a patch yet?)
Phantasy Star Portable 2
Touhou
Then there are also the N64 and GBA emulators DaedalusX64 and GPSP respectively.
These are the ones I play (minus the ones with no patch. Those are on my NAS).
My absolute favorites are Persona 3 Portable, Disgaea: Afternoon of Darkness, and Final Fantasy 7 Crisis Core. I thought Death Jr. was kinda fun too, but I don't know how popular it is.
-SRW Z2 Hakai-hen and Saisei-hen (awesome tactical RPG)
-Valkyria Chronicles 2 (good game with good if not decent story)
-Persona 3 Portable (best RPG ever imo)
-Knights in The Nightmare (difficult at times but quite enjoyable, the artworks is nice thou)
-FF Dissidia (game to be played while waiting for someone or something)
-Hot Shot Tennis (a little bit of sport for variety)
-KH BBS (really nice game, need i say more?)