1 - meeting the early versions of staples like Bahamut, Omega, etc.
II & III - I dont know anything about this games, so nothing.
IV - Kain
V - Faris and Lena finding out they're sisters.
VII - The part where Celles sings in the theater.
VII, VIII, IX, X, XI, XIV - Don't know or haven't played enough of them to pick a favorite moment.
XII - The last battle in the game, all of the battle, not just the when the player gets to fight the last Boss.
XIII - The end scenes.