I just sold my psp, no time to play it anymore so I decided to get rid of it before sony's new NGP makes prices plummet.
When I first got my psp I sort of wished I got a ds too (pokemon! yay!) but after trying a ton of games there definitely are some good ones. like RPG's a lot so that's mostly what I did with my PSP.
My all time favorite was Disgaea. The camera takes some getting used to and there's a slight learning curve but it's really awesome once you get into it. It's almost like Fire Emblem style fighting but with much more RPG elements. So there's slightly less strategy and tactics but a lot more leveling, special skills, and equipment involved. The characters are hilarious and the storyline is good too. You could play this game forever, there's so much stuff to do. The storyline is only like half of the game there's still a ton of stuff to do afterwards. One of the very few games that I played through more than once.
Monster Hunter Portable - TBH I got bored in a short amount of time but the graphics are good, combat system is one of the best I've ever seen there's actually a lot of skill. The item system is also super awesome, equipment looks cool, you can actually see armor on your char, and there's a really complex crafting system. The only problem is that you spend the whole game hunting monsters/animals and there isn't much of a storyline...
Breath of Fire III - I played this like 6 years ago but it was super awesome. One of the best storylines ever. Equipment system was decent, there are rare items but your char pretty much looks the same throughout the whole game. Grinding does help, it's not one of those games where levels are irrelevant. There's a fair amount of skill in the battle system.
YS series - There are a bunch of titles, there are some newer games with pretty nice graphics and some older more classical ones. Very simple games, I can definitely see some similarities to Zelda. There's a decent amount of skill involved (mostly just for the boss battles) but for the most part you just hack and slash. Equipment system is very simple, you don't find any rare equipment you just buy them in a store or they're given to you. There really isn't any grinding necessary.
Tales of the World - Some people say this game was repetitive and boring but I liked it... Graphics are pretty good, Item/equipment system is very in depth, and I thought the battle system was awesome as well. The battle system takes a lot of skill, there's a ton of dodging and comboing involved, it's almost on par with a game like Super Smash Bros. There's a pretty cool class system too and the storyline wasn't too bad. It's a shame they didn't translate 2 or 3. Darn Japan, keeping the good stuff for themselves.
Legend of Heroes - Super standard stereotypical RPG with a very predictable plot. I played the first one. The story is VERY long.
Dungeon Siege Throne of Agony - Another really good RPG, this one is like Diablo style (That's what I heard, I never played Diablo). Cool combat system, nice equipment system, and good storyline.
Hexyz Force - Pretty standard RPG again. Kind of cool because you can play both a female and male character. I thought the girl char was sort of cute. If you don't think so then don't bother playing.
Mana Khemia - Cool game, but the leveling system was really dumb so I stopped playing.
Valkyrie Profile Lenneth - Weird game, I couldn't get into it.
Persona 3 - Very interesting RPG. Not standard at all, it's like a combination of a visual novel and Pokemon almost. I'd recommend it though, I liked it a lot. It's hard to describe, it's like 2 styles of gameplay in one.
Class of Heroes - Don't play this game it sucks.
Star Ocean - Another pretty standard RPG imo. Not so much skill in the battle system but it's real time as opposed to turn based. Your character sprite looks the same throughout the whole game. The storyline is pretty original I guess. The game is a classic apparently and lots of people like it.
Phantasy Star Portable - Great character creation system and very good equipment/item system too. Combat is a little awkward but not too bad. Levels are somewhat repetitive and I didn't like the storyline very much.
Dante's Inferno and God of War - I'm lumping these two games into one because they're almost identical other than the storylines. You hack and slash and beat the crap out of everything and look really cool while doing it too. I managed to finish Dante's Inferno but God of War got sort of boring. Graphics are great, equipment is non existant, you really don't grind or level you just get more skills.
Assassin's Creed - I loved this one, it made me feel so neeeenja climbing up and jumping down buildings and just pwning everyone. Altair is so BA. I'm afraid of heights in real life so I guess it sort of gave me some sort of extra thrill playing this game. But really, it's a very good game for psp standards.
GTA series - I thought this game was hilarious, it's so funny just taking people's cars and running over stuff. I didn't finish it but it's definitely one of the better games out there.
Jeanne D Arc - Another strategy RPG sort of like Fire Emblem but much more in depth. I liked it a bunch. The voice overs with French accents were sort of cool.
Final Fantasy Tactics - Too pro for me, I got stuck on level 2 and gave up lol.
There are probably some games that I'm forgetting, I played through such a huge volume of them. The PSP does have a lot of great titles though. For some reason I was never a big fan of Square Enix's stuff. And wow what a wall of text.