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)