Valkyria Chronicles (although classify it as more strategy than rpg, still great game)
Persona 4
Ar tonelico Qoga: Knell of Ar Ciel (you'd really need to play the first 2 in the series first though, and they are ps2 games...haven't checked, but they might be on the psn)
Ar Nosurge Ode to an Unborn Star - Set in the Ar Tonelico univerese I guess. The Tonelico games probably aren't mandatory to play, but would be helpful. Plus, there was a prequel game that came out before, that didn't get released here. Someone wrote up a novel summary of that game, that would be helpful to read prior as well, but I doubt most people, except diehard fans, would want to go through so much trouble to play a game.
Any Tales game (although my favorite is still Vesperia, which is a xbox 360 release in NA, the other tales games are good enough)
Altelier Rorona, Totori, and Meruru (arland trilogy)- slow paced, with only just enough plot to get the game going...but, the combat and alchemy systems are quite fun to use.
Wizardry: Labyrinth of Lost Souls - pure dungeon crawler, so no plot. I enjoy those types of games occasionally.
Final Fantasy x/x-2 hd remake
Other series I like, but I'd have trouble recommending to people
Hyperdimension Neptunia series
Record of Agarest War series
White Knight Chronicles 2 (came with the first game, so only need to buy the 2nd) - multiplayer servers are shut down, but should be able to still play the story of the single player
If you don't mind downloading ps1 and ps2 classics off the psn for your ps3, then I'd also recommend....
Any of the shin megumi tensei games (although the best, imo, are the persona games)
xenogears
legend of dragoon
chrono trigger