What game I've EVER played has the most heart?
...I...I don't even think I can answer that question. So many games...So many games with real heart to them...
So instead I'll mention a somewhat recent game with a ton of heart, which absolutely made it into a stellar game:
Of Orcs and Men.
It's a somewhat low budget action/tactical RPG about an Orc (looks more like a troll, though), and a Goblin on a mission to kill the tyrannical human emperor.
I went in expecting a really short, super cheesy over the top game with little emotional impact.
What I got? I got a tightly bound, surprisingly long (24 hours, according to my steam log) game that actually pulled at my heart a little. Despite it clearly being made on a tight budget and having a few questionable design choices, it gave the world, story and gameplay EVERYTHING it had. By the end, I had stopped seeing the two main characters as an Orc and a Goblin. They had become "people". I cared about their struggle, I connected with them. I, amazingly, empathized a lot with the Orc, who at first I thought was just another meathead tough guy, but who actually is the emotional heart of the story.
Seeing a game deliver on that level...It was amazing. It knew what it wanted to do, and it went and DID IT.