As the PSP era comes to a close I think it'd be a good time to look back on all the great games it had. This list will encompass all worthwhile games, reasons for them and light descriptors. It will be slightly in order(Pirates is not really the 43rd best game), as a lot of games are dependent on the type of person who wants to play them.
I'm also choosing not to include collections like PowerStone, Japan only games like Bleach(soul carnival is pretty cool though...), remakes such as Persona, or games that are based off interesting franchises but aren't great themselves(Third Birthday, Bloodlines) I'm also not including games I haven't played such as Tales of Vs.
Honorable mentions:
Naruto series(strangely good controls on the 2D fighters released in the US)
Ys(check the DS games as well if you're interested)
Kenka Bancho: Badass Rumble(There's a great QL on it, and despite it not being one of the best it's one of my favorites)
Star Wars Battlefront Elite Squadron(Cool tech, poor design)
GTA Chinatown Wars(Remake)
Hammerin' Hero(Great look, Too Simple)
Knights in The Nightmare(Also a DS remake)
Riviera: The Promised Land(GBA remake)
Phantasy Star Portable(Mediocre game, Some cool ideas)
Exit(Worth a try, but not one of the best)
Space Invaders Extreme(DS version just as good, but a great PSP game)
Downstream Panic(Great Puzzle game)
Gran Turismo(It exists, and it's what you'd expect)
Harvey Birdman: Attorney at Law(It's really part Phoenix Wright, part VN. If you want some comedy pick it up for this or the wii)
Medal of Honor: Heroes 2(I hated it, everyone I knew loved it. I hate enough to keep it on the list, but not enough to take away what people might like)
Thrillville(Combination Minigame collection, and light Theme Park Sim. It has a weird charm about it, take that as you will)
Powerstone Collection(Still great)
Every Extend Extra(Weird shooter, interesting mechanics)
Killzone: Liberation(Isometric 3D shooter. Well reviewed, but personally not great)
WTF: Work Time Fun(Minigame Collection.The first minigame is capping pens, so it's not for everyone)
Me and My Katamari(More Katamari. If you want a portable version, wait for Vita. If you just want more then try this.)
Tokobot(Fun, but not great)
Metal Gear Acid(Not my thing, but a lot of people love it. There's also a sequel)
SSX: On Tour(I've never played an SSX title, but otherwise from what I've heard it'd be on the list)
Shadow of Destiny(Weird remake of a story focused game with time travel mechanics. It's interesting as not many games are similar)
Ace Combat: Joint Assualt(If you like Ace Combat it exists, however I don't think it's one of the best on the PSP)
Blazblue(Some love it, some hate it. Either way the control is unique and it's imperfect)
Darkstalkers(Again, good for people who haven't played the series)
Mercury Meltdown(The original is on the PSP but this evolved so much more. It's still not one of the best but what can you do)
Manhunt 2(It's suprisingly well designed for a PSP version, but other than the unique premise and polish, it's not one of the best)
Phantom Brave(I really must play this after what I've heard)
R-Type Command(Interesting sequel that never lived up to it's past)
Parappa The Rapper(Not just a port, but also just not as good as it was. Maybe it's just a bad port)
Prince of Persia 2/3(The lesser of the three, but still worth a run if you missed them)
Metal Slug Anthology/XX(Both maintain the look and feel, but Metal Slug isn't as good outside of the arcades)
Samurai Shodown Anthology(I'm not great at fighters but this seems like a well designed series. Having said that, I could be totally off and deserving of a beat down)
Persona 1/2/3(I've never beaten the Persona series, but I've seen both it's history and it's quality enough to add these ports to the list)
Tactics Ogre: Let us Cling together(Remake)
Tekken 6(Still intricately designed, but never reached the height of 5)
Final Fantasy Tactics(Not my game, but beloved by many)
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Valkyria Chronicles is a great way to get both strategy and shooting for the PSP(though more of the first), it's a hybrid RPG/Squad management game. Overall it's a little long, and is extremely tough at points, but the gameplay alone makes it worthwhile.
One of the absolute best games on the PSP, Crisis Core melds a great fighting system with RPG basics and an interesting chance mechanic. Not much bad I can say other than the cheapness of some of the boss battles.
Patapon has gotten kind of old by now. Not to diminish it's appeal, but you should really only play two of the games because the mechanics don't last forever. That being said it's wonderfully innovative in it's music based Group control combat. That beat keeps it working way longer than it should, and when you put the RPG mechanics on top of that it's one of the best on the PSP.
Considered by many one of the best if not the best game on the PSP, Daxter expands the Jak and Daxter storyline while giving you a small open world, plenty of mission variety and slow combat upgrading. It's a detailed and well designed game that is often the face of the PSP.
I'm not much of a fighter game guy, but for some reason I got hooked on this game. Maybe it's the amazing graphics or maybe it's that it doesn't need that second analog stick. Either way I played the hell out of this game and never looked back.
This...isn't really like any other MGS game out there. It's more of a pokemon collection/squad management game combined with MGS. It's got some neat mechanics, but the objectives and story really pull it all together. Don't play Portable Ops Plus. It's nearly the same, but without the goals it really loses the fun.
I love Level-5. Guys that made Layton? Dark Cloud? White Knight? This is just as good. Better depending on your tastes. It's got a great art style both in 3D and 2D, as well as classic strategy tile based combat and a fun if imperfect story.
When I first played Just Cause 2, I thought the Stunt jumping was Unique. Actually, it came from this series. Styled like a cartoon/action cop show Pursuit Force made car combat(not twisted metal style) a thing. It's stupid, fun and it never really got the attention it deserved.
Like most of the best console inspired games on the PSP, this was designed to be a greater experience. It allows for a crazy amount of team control while still retaining a shooter feel all while fitting into the PSP controls. If you like shooters you should check this out.
Who cares about HD. Sure you can buy this on the PS3, but it really shines when you play it on the bus, or honestly in the bathroom. This top down musical shooter is fun, beautiful and symphonic. One more thing, you should wear headhones when playing.