I never got to the end :( maybe more people didn't make it either?
I remember being stuck somewhere deep underground with pyramid head.

Bioshock 2.

Final Fantasy X. The while story with Tidus and Yuna was so sad. The end was especially sad. The only other game that comes close is Tetris.

Final Fantasy VII and Final Fantasy X.

Kingdom Hearts.

Metal Gear Solid 3 and 4.

Heavy Rain.

For me its an easy choice... I'd have to go with Heavy Rain & Red Dead Redemption... Heavy Rain from start to finish, insanely emotional in the middle to the end. and RDR had maybe one of the most emotional endings... especially if you actually did that "strangers quest" after and played as his kid and killed the guy who was the reason your dad died. Powerful stuff.

Lunar Silver Star Story - http://en.wikipedia.org/wiki/Lunar:_Silver_Star_Story_Complete
I think an under-rated RPG and far better than a lot of bigger named ones.

Heavy Rain - Just engaging storyline - closest to a movie feeling
Flower - One of the most peaceful, engaging experiences I've ever had with a video game. Loved every second of it.
Metal Gear Solid - Really engrosses you, if you let it. Some of the most engaging games I've ever played. Loved MGS2.

Chrono Cross, MGS 3 and 4, TLOZ Link's Awakening, that's all I can think of at the moment.

Shadow of the Colossus

Final Fantasy VIII
Final Fantasy X
Kingdom Hearts

Mass Effect Trilogy : especially the "good" resolution of the Tuchanka and Quarian Missions in ME3.
Walking dead : I usually always make the most useful, pragmatic choices in games like these, but for some reason, i couldnt in this, because of Clementine. Its ridiculous how attached you get.
But the most emotion I have felt during a game, since hunger for revenge probably counts as one, is Max Payne. Played through in one go, and the final level was just sweet,sweet satisfaction.