One of my first memories was sitting with an uncle who died when I was nine, playing final fantasy 1 and having him read the scrolling text along with me when you cross the first bridge and really start the game. Whenever I play through that again now, in any incarnation it gets me somewhat emotional.
From a game itself though... Hm. Final Fantasy 2(4) had me real upset when Polom and Porum died, though I don't think I actually cried over it. Final Fantasy 3(6) did a similar thing with Cyan finding his family dead.
Very few early games tried to touch on emotion like that which is why I think I jump to just Final Fantasy games. Since then, there have been a lot that really tried to touch a nerve with you, but as I got older I could appreciate the drama of the moment without it really upsetting me.