This is so subjective, and a lot of good examples - (including USA vs Russia in Lake Placid).
In baseball - 1986 NLCS and WS Game 6's. NLCS Game Six was crazy - with the mets winning in the 16th inning. (The mets of that team have often said that they would have lost a game 7 to Mike Scott). Obviously the Bill Buckner story - when I talk to most people about it, they don't realize that the game was already tied at that point, and that the Red Sox blew a 5-3 lead with two outs in the ninth. That game was also on simchas torah during hakafos, and our shul closed the street down for dancing outside. But no one was dancing, the policeman in his little scooter had the radio on, and we were all crowded around to hear the game.
In hockey - the Easter Epic - Game 7 of the first round between the Isles and Caps which ended almost 9 minutes into the 4th Overtime (so they literally played more than two whole games).
As a Giants Fan - my favorite superbowl was the 2008 win over Brady and the Pats - including the David Tyree "Helmet catch"
Obviously, these make me a bit of a homer, but I would say the best game in recent memory where I wasn't a fan of either team - was Game 7 of the 2016 World Series between the Cubs and the Indians. The Rajai Davis Heroics (especially the homer off of chapman), the rain delay, I stayed up for all of it, even though it ended after midnight on the east coast. (Sorry for the outcome Dan).