I owned a psp a few years ago before selling it, but before I did I mostly used to pick up really cheap strange games. So I got into the habit of playing things that usually sucked but were interesting in a weird way.

Stuff like
Harvey Birdman
Brooktown High
WTF
Gitaroo Man (although that didn't suck at all)
Sims 2 (totally different to regular sims games)
Futuristic Harvest Moon
Echochrone

I just picked up another psp for 50 bucks and I was wondering if there were any other titles in that kind of sucky-but-interesting genre that I'd missed out on in the past few years. It's easy to find great games, but these kinds of titles are usually buried deep. Thanks
 

I hear Half-Minute Hero is good.

Parappa the Rapper 
 
Valkyrie Profile is weird in it's own way too