It depends really by in what way you mean emotional. I've played games with very little story but they've been so magical that I felt sad when I completed the game as I wanted it to go on forever.
For me such a game would be Super Mario World and Mario 64.
There have been many games I've played with characters I empathise with such as Shemue and games which to me felt truly epic like Panzer Dragoon Saga and even games which I kind of agreed with me in a political way such as Grandia 2 (humanism).
This gen though the two game that most moved me I would say is Lost Odyssey and most recently The Journey.
Lost Odyssey because of the scene with the children crying over the death of their mother. The facial animation and the voice acting was amazing. The lead characters 'Lost Memories' also provided a scope for exploring different human sensibilities.
The Journey because of the striking visuals, cinematic camera angles and soul stirring music which with the journey of the protagonist just stirred something in me on a deep artistic and emotional level.
