[Doctor]

The Aeris death scene is a popular choice, I'd say.

But I think that the one scene in a FF game that made me cry genuine tears was during the ending in FFX, when Yuna, who sees that Tidus is slowly fading away and is leaving, desperately tries to hug him, but just falls right through him. That particular scene gave me a feeling that I've experienced before; one of no longer being able to be close to someone dear to you. It's an emotionally overwhelming feeling.