I will echo a few others in the thread and say Walking Dead Season 1 got me the closest to full on crying. I have teared up in a few instances throughout gaming:
FFVI Opera House scene. My favorite scene in all of gaming. Hit all the right notes for me at a young age.
Bioshock Infinite's ending
Batman Arkham Knight had me tearing up with nostalgia toward the end when Conroy did the iconic "I am vengeance ..." line.