Despite a slow start oh those many years ago, there are now a lot of games available for the PSP. Some of those games are really good and some are best avoided unless you have more money and time than any gamer has ever had before. So I'm here to help. The following action-adventure titles are games that I played and enjoyed. They're not all the best game ever, but they are--in my obviously biased opinion--worth playing. But then, I'll try almost any action-adventure game at least once . . . (Oh, and I will edit this list from time to time, so check back.)
Daxter
Sony
This handheld-sized installment in the Jak and Daxter series features the eponymous smaller half of the team and packs in just as much fun as its PS2 siblings. I gave the game 4 1/2 out of 5 stars, for what that's worth, and stuck with it long enough to find every Precursor orb and get gold on all the minigames. I even bought the PSP-2000 Daxter bundle and gave my extra copy (of the game, not the PSP) to a friend. He liked it too.
Gurumin: A Monstrous Adventure
Mastiff
For some, it was hard to overlook that the hero in Gurumin was a cute little girl. I really hope that didn't dissuade too many players, because it's really a fantastic game. Technically, it's categorized as "action-rpg," but it's not out of place on an action-adventure game list. And besides, I really like the game and it's my list.
LEGO Indiana Jones: The Original Adventures
LucasArts
I could probably have put any of the LEGO games on here, as they're all pretty darn good. But I have a soft spot for Indy, and it's the only LEGO game I actually got around to reviewing. But if you prefer Star Wars, then by all means pretend the heading says "LEGO Star Wars." I admit I was dubious way back when the LEGO games were first announced, but I've since realized that LEGO + [your favorite movie] = so very much fun.
Ratchet & Clank: Size Matters
Sony
Here's another PS2 duo that made it big on the (really) small screen. Size Matters is really only small in the sense that it fits on a PSP. Otherwise, it's just as good as the other games in the series. And hey, it even got ported to PS2, so you can play it on your TV if you prefer.
Dead Head Fred
D3Publisher
This game won an award for its story, so if you like games with story (which I do) Dead Head Fred is a good pick. Plus head-swapping is just fun. I think this was the first game I played on a PSP Slim and the first I played on my TV via video out cables, which could bias my opinion a little. But I don't think so.
Death, Jr
Konami
This might seem an odd pick for a "best" list, as I don't think too many people thought Death, Jr was really all that great. Maybe I just have a soft spot for it since it was one of the first games I reviewed for this site (I gave it 3 1/2 stars). But really, I think it was the cool visuals and the morbid sense of humor that endears Death, Jr to me. I haven't played the sequel yet, but I fully intend to once my hectic schedule settles down a little.
Tomb Raider Legend
Tomb Raider Legend made the franchise fabulous again. I have to admit that I never made it all the way through Tomb Raider III on PS2, and wasn't tempted to pick up another one until Legend. I liked it so much I'm actually tempted to go back and play the PS2 games I missed between TRIII and Legend, and maybe even finish III.
Secret Agent Clank
I haven't actually finished this game yet, and have had to set it aside to finish and review a couple other games, but already I love it. I think Secret Agent Clank even manages to be better than Size Matters, which is saying a lot. When I do get around to the review (oh, and I will), I'll probably give it 4 1/2 stars, which is (I think) the highest score I've ever given a game. But I'll have to play it all the way through to make sure.