What's some great overpowered equipment to nab really early in the game?
You can steal Karkata from Trial mode's third trial (from the main cactuar, can't remember its name). 1H Sword with 92 ATK, inflicts Confuse on hit. It doesn't really get better than that for the first 20 hours.
Some chests also have some solid gear to get, thought nothing exactly game-breaking, I think. You can find maps of chests and their %s (chance to appear, chance to have an item, etc.) here http://www.fftogether.com/forum/index.php?topic=2764.0
Best treasure so far has probably been Main Gauche, which you can get in the Ozmone Plains during rain. It's a dagger that adds 50 to your evasion.
e: You can also steal the Sword of Kings from Trial 10's boss. 2H Sword with 53 ATK, doesn't need a license so anyone can equip it.