The PSP is a beast with a glorious library. I own three PSP 1000s, so yeah, I love the handheld fervently to say the least. It's a JRPG & SRPG goldmine, but there's plenty of nice action adventure and platformers as well. I haven't played every PSP game I own yet by a mile, but the very best ones I've actually finished so far have been:
Valkyria Chronicles II (fantastic SRPG with tons of fun missions)
The 3rd Birthday (terrible story but great gameplay and amazing graphics)
Brave Story: New Traveler (very solid JRPG with excellent pacing)
Final Fantasy VII: Crisis Core (fun action-RPG with the best ending of any FF I've played)
Resistance: Retribution (kick ass shooter with nice controls)
Metal Gear Solid Portable (underrated and feels legit)
Metal Gear Solid Peace Walker (crazy fun and very deep gameplay)
Metal Gear Ac!d 1 & 2 (loved the hell out of these CCG/SRPG hybrids)
Ys Seven (do not skip this one)
I'm probably forgetting some. The PSP library is gargantuan so that's not hard to do. If you are into SRPGs do not miss Jeane d'Arc or Tactics Ogre: Let Us Cling Together.
I know you're into visual novels & adventure games OP, here's what's official for USA PSP:
Corpse Party & Corpse Party Book of Shadows
Disgaea Infinite
Fate/Extra
Hakuoki: Demon of the Fleeting Blossom
Shadow of Destiny (AKA Shadow of Memories)
Sweet Fuse: At Your Side
There are quite a few English fan trans games available on PSP. Probably the most popular are:
Dangan-ronpa
Final Fantasy Type-0
Valkyria Chronicles III