I still think it would be cool if there were chests that you could just find and open in the world that were unique to your character, even if the contents of the chests were always the same each time, and you could only open them once per character.

The completionist in me would be very excited about treasure chests like that. They could even base some achievements off of finding all the chests hidden in the world.

Remember Chocographs?

Something like that (or exactly that, since the game has plenty of FF fanservice) would be a fantastic exploration-based quest in this game.