Let's see, if we're going purely off of story then it would be:
Lunar: The Silver Star (Kind of a tie here between the original Sega CD version and the last PSP remake, there are elements of both stories that I really liked, but I feel like they really toned down the game's story when they first ported it. It's not as dark as it was on the Sega CD, but being able to keep Luna and actually seeing some scenes from what might have been, but absolutely never will be Lunar 3? That almost makes up for it.)
Lunar: The Eternal Blue (No question here, the original Sega CD version wins hands down.)
Shining Force II
Phantasy Star III
Phantasy Star IV
Final Fantasy IV
Final Fantasy VI
Shining Force CD
Grandia II
Grandia
Chrono Trigger
Suikoden II
. . Really all the Suikoden games so I, III, IV, and V, too.
Tactics Ogre
Final Fantasy Tactics
Here's one out of left field, wonder if anyone else even knows it: Beyond the Beyond
Ar Tonelico I
Ar Tonelico II (to a lesser extent, since it's just story it makes this list, but I absolutely detested the change they made to the battle system)
Legend of Heroes: Trails in the Sky
I should probably just stop here, I could keep going on for quite a while, these are in no particular order, and just because I left off a game doesn't mean I don't love it just as much. Just means I had to stop listing games and go do something slightly more productive for a bit.