In the hunt for games

[TkSilver]

Well the psp probably had one of the best (at least for the time) collection of action/adventure rpgs with some ports of some heavy hitters and some original series.

Teleportation to a field to kill (dinosaurs, aliens, or monsters) type games. (More action little less rpg)
Personally recommended titles: phantasy star portable 2 and god eater burst.
Monster hunter series
Phantasy star portable 1/2
God eater series (only burst has an official translation)

Squaresoft action rpg titles (squaresoft story telling with a more active approach to combat then most final fantasy titles)
Personally recommended titles: All are unique to each other and recommended to try and see if you like their battle systems
Final fantasy crisis core
Kingdom hearts bbs
Final fantasy type-0

Diablo style loot fests
Recomended if you like these types of games but do not care about the story that much.
Untold legends series

There are others like kingdom of heaven and grumin. Probably many many more I have not listed either due to forgetfullness or being hard to classify (like silverfall - a port from a windows game that tried to be kinda like sacred but not really)