If I had my way, I'd put pretty much the entirety of To The Moon in this list. It's one of the most beautiful and moving games ever created, and it's filled with dialog that taps into the ordinariness of everyday conversations, while simultaneously making each one feel special. Also.... that music.
But the scene where Johnny and River meet for the first time on a park bench and talk about the origin of their names, the stars, and the carnival prize John won -- which he gifts to River -- is just wonderful. With echoes of the equally lovely Before Sunrise, it's capped off by the pair arranging to meet at the same time, in the same place, the following year. Tissues at the ready.