In no particular order:

1. Half minute hero: quirky, quick, hilarious, and a bit stressful at times. It's a perfect mobile title.

2. Disgaea: infinitely scaling/playable strategy rpg. You can power through the game pretty quickly, or take your time and slowly become a god so strong that even friendly healing spells miss you.

3. Patapon (2 and 3): rhythm based game about using drum commands to get your little dudes to complete missions.

4. Dj max (any/all):dance dance revolution for your thumbs. Korean game featuring Korean artists. One of my favorite continuing series.

5. Monster hunter (freedom through portable 3rd): action rpg all about them boss fights. Kill monsters, use their parts to hunt tougher monsters in a game all about incremental progression, inventory management, and learning movement patterns to git gud.

6. Patchwork hero: do you like the dig dug stages where you slice off bits of the land to drop enemies into the water? This is that except they are airships and you must defend your homeland

7. Lumines (1 and 2): musical game about matching colors and making combos in a tetris-esque style game. The devs went on to make tetris effect.

8. Cladun 2: so I heard you like action rpgs, infinite scaling, pixel art, and enough grindy systems that even my power tools get jealous. Enter Cladun.

I'm wondering if I missed any I spent a disproportionate amount of time on.. But these were the first ones to pop into my head.

Edit: I am finally home so I can look over my games... and I definitely forgot a couple.

1. LOCO ROCO! It's a very simple platformer in which you guide your little dudes through stages by using the L and R to tilt the stage. You collect fruits to increase your dude into a larger dude, or several dudes, depending on the situation.

2. N+ Be a platforming Ninja, die a lot.. like a lot a lot. It's.. simple and hard and so satisfying to clear one set of 4 stages at a time. You can even make your own stages if you're feeling up to the challenge.

METAL GEAR SOLID: PEACE WALKER

The PSP was a fantastic system for racing games, despite the lack of an analog throttle. If you want to get into this genre, this console is a great starting point:

• Outrun 2006 Coast 2 Coast: Perhaps the best version of Outrun to date and an ideal game for people not that used to the genre. Easy to learn, hard to master. It's gorgeous to look at, the arcade-style gameplay loop is perfect for a portable system and so much fun that you don't really mind if you lose.

• FlatOut: Head On: A very different take on the arcade racing genre. This one blends challenging stock car racing with over the top minigames that involve drivers being used as virtual bowling balls. It all makes sense, just try it out, if you don't mind the generally high difficulty. Head On is a near 1:1 port of Flatout 2/Ultimate Carnage, just slightly simplified visually to run on the system.

• Colin McRae Rally 2005 Plus: As the name implies, this is an ever so slightly reworked port of the PC and console game. It's near 1:1, with only some modest visual downgrades. This is where Codemasters' rally games peaked prior to Dirt Rally - and unlike Dirt Rally, which is a bonafide racing horror game, this title is accessible to players of all skill levels, with only players who are really into simulations being miffed about the lack of per-tire simulation. Career progression from modest FWD cars to spicy Group B monsters is absolutely exemplary and tracks are a delight, both visually and in terms of design.

• WipeOut Pure: This might just be the best game in the series. Perfect controls, perfect track design, perfect career mode. The only flaw is that on original hardware, the frame rate isn't always stable. I would highly recommend Pure as an entry into the series.

• MotorStorm: Arctic Edge: One of those "concession games" for older systems as the new generation was already out, releasing for PS2 and PSP - but it's actually more fun than the mainline games, at least in my opinion. It makes perfect use of the hardware, the vastly different vehicle classes and their strengths and weaknesses are tons of fun, just like in the mainline series, and the track design is wonderfully vertical. It also looks magnificent, which seems to be a trend among PSP racing games, now that I think of it. The developers were so confident in the tech of this title that they included a photo mode, which I don't think all that many PSP games have.

• Gran Turismo: Everyone expected this to be a mobile port of Gran Turismo 4, but it's not. While it retains the presentation, large roster of licensed cars, real-world and fictional tracks and highly refined simcade driving physics, there is no conventional campaign. Instead, you create your own racing events and can only buy from a small selection of cars each virtual day. It's an interesting concept. Do try it out just for the wow-factor of having a game this photorealistic on the PSP and perhaps stay for the fun you can create yourself. If you lack the creativity, people have created random event generators: https://sites.google.com/site/gtpspcampaigngenerator/ Gran Turismo is also to this day - and someone correct me if I'm wrong - the only mobile racing game that features a full 1:1 recreation of the iconic Nürburging, which alone makes it worth playing.

• Midnight Club 3: Dub Edition: Classic Midnight Club on the go, but this time with customization. What's not to love? Controls are so accurate, every other arcade racer will feel sluggish by comparison, it looks great (again), the cities are huge, there's tons to do, it has just the right difficulty, challenging without being unfair and it perfectly encompasses this bygone era of early to mid 2000s street racing, even better than any Need for Speed of the time, in my opinion. Like all of the Midnight Club games, this one is also great for just randomly driving around without any goals in mind.

• Test Drive Unlimited: Speaking of randomly driving around, this game represents perhaps the very peak of this idea. Shrinking down the entire island of Oahu at 1:1 scale with its real-world street layout (and some creative liberty in regards to buildings), this one is truly a miracle of optimization. The result may not be the prettiest racing game on the system, but the bouncy, responsive driving physics - which so closely mimic that of Gran Turismo 2 that I'm convinced they straight-up copied them - more than make up for any visual shortcomings. This also means that this cut down version of the 360/PS3/PC game (visuals, some vehicles and customization) is more fun to play than the big version, which has comparatively flat driving physics that straddle the line between arcade and simulation nowhere near as well. If you like the idea of living the idea of living the life on a Hawaiian island, buying homes and cars to fill their garages with, racing down long coastal straights and maneuvering through twisty mountain roads, then this might be right up your alley.

Patapon

Final Fantasy Tactics: The War of the Lions is an obvious one.

Daxter and MediEvil Resurrection are two of my favorites.

Weird that nobody mentioned the GTA games on PSP

Vice City Stories and Liberty City Stories are absolutely awesome GTA games somehow squeezed onto the PSP

Making a new comment off my other reply because I have more niche recommendations: Carnage Hearts EXA, you program robots to fight each other, actually quite in depth.

Good RPGs: Valkyrie Profile Lenneth, Legend of Mana (originally PS1)

Phantasy Star Portable 2: pretty decent PSP version of the phantasy Star online game.

Dissidia 012 Final Fantasy, kind of a weird "fighting" game, very unique mechanics, but really fun, lots of fan service, and tons of really really excellent music from across the final fantasy series

SoulCalibur Broken Destiny: solid entry in the series even on portable

And Castlevania Symphony of the Night (originally PS1)

Prinny: Can I Really Be the Hero?

It's not amazing, but as far as side-scrolling platformers go, it was fairly compelling.

Also the PSP can play PS1 games, so you have access to a pretty big library of classics.

The Parasite Eve game on the PSP was pretty good. Mega Man Maverick Hunter X and Mega Man Powered Up were both good too.

I don't know if I could call it good, exactly, but one unique concept that I haven't really seen captured anywhere else was the Dungeon Maker series on PSP, that allowed you to build dungeons that you would then explore/fight/loot, to give yourself funds to build out further/deeper, ad infinitum. It was clunky, controlled pretty stiffly and basic as ARPGs go, and after a certain point you kind of went on autopilot, but there's a certain je ne sais quoi to it that I really quite enjoyed, especially if you planned out your builds. I think a similar title was released on the DS but it was turn-based and not particularly well-executed.

Every PSP I own must have the following games ready to go at any time:
-Lumines
-Rock Band Unplugged
-Patapon 2
-WipEout Pulse
-Final Fantasy: Crystal Defenders (gorgeous small tower defense game, but with enough strategy behind it for hours of play)

metal gear portable ops 😎

My personal must-play PSP titles:

• Monster Hunter Freedom Unite
• Patapon 2/3
• Lumines
• Wipeout Pure
• Gran Turismo
• MGS: Peace Walker
• Tekken 6

Are action RPGs okay? If so, Ys: Oath in Felghana. It's a remake of Ys III, which is also available on other platforms. However! The PSP version has features that other platforms do not, the main two being voice acting (other ports just have beeps when text is displayed) and a whole new mechanic, which is reminiscent of Final Fantasy's Limit Breaks.

One potential caveat, though: this is probably the hardest entry in the series. Fights are hard, but never cheap. If you're okay with that, you should have a good time.

Maybe not your speed genre-wise, but Final Fantasy Tactics has a version for the PSP. Goated game.

Jeanne D'Arc and the PSP's Twisted Metal were damn good.

If you like ARPGs, I remember enjoying Untold Legends a lot as a kid. Was a PSP launch title, and had a very dark themeing. I don't know if my kid memory can be trusted on the quality but I've seen a few other people recommend it on here so I doubt it's bad.

god of war (2 games)

I’ve got to recommend Monster Hunter Freedom Unite. One of the most legendary games in the series for its sheer amount of content. It might feel a bit dated if you’ve played World or Rise, but still good

The Armored Core Portable games are really good, if you can live with thr wonky controls. There's Tekken 5 Dark Ressurrection, great port of the same.fighting game of the PS3. Gran Tourismo Portable is nice, alongside Wipeout Pure and Pulse, and Ridge Racer are all nice racing game options. And I cannot go without mentioning Monster Hunter Portable 3rd! One of the best in the series, easy to get into and has a fan translation available.

I hope I don't derail the thread, but I feel like this is an opportunity to ask about a couple of games myself:

How are the two PSP Valkyria Chronicles games and do they perform well on PPSSPP? Preferably Android (Snapdragon 670, Adreno 615, if that helps), but PC is fine as well.

I really enjoyed the first on PC and bought IV ages ago, but I haven't played it because I kind of want to play the others first (which were never ported to PC or any other console for whatever reason).

From what I understand, all the games are more or less self-contained, so I can play IV without missing much but I'd really like to give II and III a shot if I can.

Star Ocean Second Evolution for me, faithful port of the original PSX RPG. It's very good, but if you do play it, it is worth researching which two skills help you power through the game lol.

If you're into racing games, I would suggest ATV Offroad Fury Pro. I've been thinking about it a lot lately; it's one of my favourites for that system

EDIT: Also, Monster Hunter games are a classic! I've never played any for the PSP, though

Coded Arms and its sequel are surprisingly competent shooters on psp. Highly recommend.

Maybe try Phantasy Star Portable 2

• Tekken 5: Dark Resurrection and Street Fighter Alpha 3 Max are in top shape on the PSP with extra characters and decent single player content.

• Rock Band Unplugged is a stealth sequel to Frequency and Amplitude, worth a try for juggling a whole band's worth of instruments.

• I also had a lot of fun with Half-Minute Hero, Lumines, Everybody's Golf Portable 2/Hot Shots Golf Open Tee 2, and the racing games DdCno1 mentioned.

• And that's not even getting into the excellent options for emulated games, both retail and homebrew.

• RetroAcheivements PSP games sorted by number of players is a good way to see the most popular games.

• Mana Khemia: Student Alliance (really one of my highest recommendations, not just even on PSP. The best "magic school" game out there, imho)
• Metal Gear AC!D
• Patapon
• Pixel Junk: Monsters
• Neverland Card Battles
• Monster Hunter Freedom Unite
• Metal Gear Solid: Peace Walker
• Aegis Eclipse: Generation of Chaos
• PoPoLoCrois
• Dynasty Warriors 2
• Lunar: Silver Star Story

Honestly, I could do an entire separate comment of just RPGs, there were SO many good ones for PSP.

Syphon Filter Dark Mirror and *Logan's Shadow.. ....*if you don't mind unfulfilled endings and enjoy the gameplay

God Eater

Ghost in the Shell: The standalone Complex (play with vita or emulation for dual stick controls)

Coded Arms (play on vita or emulation to have dual controls)

Super Stardust Portable

Pursuit Force and it's sequel Extreme Justice

Crisis Core is the only PSP game I've played. It's pretty good. I suppose Monster Hunter is worth trying out, too.