It's hard to choose, but I'd say Elfen Lied.

I know it's old but Fushigi Yugi made me cry the most.

List of touching animes
1) Fruit Basket
2) One Piece
3) Bokura Ga Ita
3) a lot of more...

Rave master.....rave master....

Fullmetal Alchemist comes to mind..It did make me cry,a number of times.

I would say Sword Art Online. The middle part when they logged out was to sad~ TOT

Attack on titan