Partly a question of what you mean by "The FF experience", but I choose to interpret the question as what qualities I hope to see in future Final Fantasy games. Here are a few such:

Character development. This is one of the things which make me less fond of I-III than later games. I want to see as many characters as possibly be affected in one way or another by the events that unfold throughout the game. The main characters in particular, but minor characters as well.

Twists of fate. I like it when unexpected things happen and when every event is the result of tons of background story. I dislike it when the main party is decided from the start and/or "fated to be the light warriors" or whatever. I'd rather see them brought together by a series of events, since this makes them more human and easier to identify with.

Exploration. I don't want to be forced to strictly follow the storyline. I want to be able to take an in-game break and explore the world (preferably a big one), get the bigger picture, do side-quests and minigames. This ties in with wanting to feel free. The lack of a world map in X[-2] puts me off a bit.

Treasures/Secrets. I want to be properly rewarded for taking my time with the game, for deviating from the storyline and taking the time and effort to go play that minigame for a while or to check out that strange-looking wall over there.