Terranigma. In pretty much all games you go through a LOT of stuff. Relationship drama, battles, dilemmas etc. But for me, Terranigma really hammered the weight of all that you accomplished down.
At the end, you return back to Crysta and face Gaia. There's a weird mix of futility and progress that I felt there, but what really tore me up was the day you were given at the end. That's just brutal. A lot of games acknowledge the end of the game with the credits, some do more, but rarely do we see bittersweet endings. Terranigmas ending was way more bitter than sweet. It stared you straight in the face and said, "This is it. When Ark goes to sleep, this story is 'over'."
To not only have a proper, mostly bitter, ending, as well as acknowledging the end of the game in ingame terms, is to me a stroke of genius.
Both Terranigma and Illusion of Gaia were very moving to me throughout. Especially:
in Illusion of Gaia when that pig sacrifices itself to feed the starving tribe, or in Terranigma when that goat practically forces you to eat it once it's dead so you can survive Link's Awakening moved me more than most other games. And more recently The Talos principle and more-so its expansion, The Road to Gehenna have been really emotional experiences at times. (Gehenna's ending, though short was really amazing).