So I just bought a PSP...

[Ghostkai]

Patapon 2
Monster Hunter: Freedom Unite
Metal Gear Solid: Peace Walker (seriously, every post should have suggested that.)
Dissidia 2
Crisis Core
God of War (both of them)


At the very least, get two of those titles. Peace Walker is amazing (playing through it again atm with my girlfriend.)

Once you have a couple of good titles, you'll adore your PSP. Trust me.

 

[daftnoize]

Half minute hero definately. Amazing game

 

[mireko]

Wipeout: Pure is great. That's the only non-JRPG I can recommend that hasn't been mentioned yet.

Unless you're okay with SRPGs, that is. If that's the case:

Tactics Ogre: Let Us Cling Together
Final Fantasy Tactics: The War of the Lions
Valkyria Chronicles II
Jeanne d'Arc
Disgaea: Afternoon of Darkness
Disgaea 2: Dark Hero Days

 

[Horizontalvertigo]

Patapon. Definitely.

 

[dimensional]

The only 3 games I have ever really played on the psp to any serious extent without getting bored are:-

Crisis core - short story but missions are good if a little tedious at times and last secret boss is a beast unless you abuse the system (like me).
Dissidia duodecim:- loved this game clocked over 100 hours much better than the original still go back to it occasionally although it does take a while to get used to.
K-ON! Houkago Tea Time:- admittedly im a fan but as ive spent over 200 hours on it so far and no one else seems to have heard of it ill mention it here,its a solid rhythm action game but since its import only if you dont know/like the anime its based on dont bother getting it gitaroo man is also a fun locally available rhythm action game if you like that sort of thing.

The only one I can think of but havent had a chance to try yet is Valkyria Chronicles 2 because the first one is awesome (but on ps3)

 

[munx13]

Both God of War games
Patapon 1/2/3
Resistance: Retribution
Motorstorm: AE
Burnout Legends/Dominator
Killzone: Liberation
Wipeout Pure/pulse
Medal of Honor: Heroes 1/2
Both Ace Combat games

 

[Hound174]

Kingdom hearts: Birth by sleep.
Higurashi:daybreak (ebay)
Tekken dark ressurection.

 

[cainx10a]

1. Gundam Battle Universe -Import- (Best Gundam Game ... Ever ... the language barrier is non-existent with the help of a translation guide) ... Playing its sequel GAS is a nightmare since you need a translation guide at hand, often, always.

2. Gundam vs Gundam Next Plus -Import- (The language barrier is not in the way. Best Fighting game with Gundams, ever).

3. MGS: Peace Walker (From what reviewers have said, it's more packed with content than even the console releases)

4. Final Fantasy: Dissidia 012 (It even contains the campaign for the first game. How cool is that?)

5. Syphon Filter Series

6. Disgaea 1/2

7. Jeanne D'arc

You will find the amount of good games on the PSP to be amazing and ever surpass that of the DS in terms of home console level of quality and content.

I wanted to add "Monster Hunter" to the list, but I suck so bad at this game.

 

[Tickolas]

1. Patapon 1/2/3.
Many, many hours of puzzle/strategy/rhythm/rpg gameplay. Play them in order or you'll never want to go back to the previous one, they evolve exponentially =)
2. Star Ocean 1/2.
Great remakes superior in every way to the original SNES/PS1 versions. Plus Pericci is cute ^^ (http://tickolas.deviantart.com/#/d2gij8b)
3. Kingdom Hearts: Birth By Sleep
It's Kingdom Hearts, good action with three separate storylines. Worth checking out if you've played and liked previous titles.

 

[Pingieking]

If you like Metal Gear, Peace Walker is an absolutely fantastic game.

This. Best Metal Gear game ever.

Why hasn't anyone said Persona 3 Portable? It's only like, one of the greatest JRPGs ever made. Persona 1 and 2 are also on the PSP, but they're not as good as the P3P.

Valkyria Chronicles 2 and 3 are also highly recommended. Tactics Ogre, FF Tactics, Jeanne D'arc, are all great tactics RPGs. The two Disgaea games are great.

Patapon is great and will drive your friends insane.

If you can read Japanese, then the library gets crazy awesome. There are so many great Japanese games that never made it out of those islands.