The greatest game I've ever seen was a soccer game that my team played. Grêmio x Nautico.
The competition wasn't big, it was the finals of the Serie B of the national championship (division 2). It had a lot at stake though, lose and the team doesn't go back to the serie A, and with tons of debt, the team was in jeopardy of ceasing to exist.
It was an away game, in a very loud crowd, with a very dirty organization. They had just painted the lockerroom (!) before the game and sealed all ventilations, so the paint smell would stay. There was no hot water and no bathtubs, all broken.
The game went with parity in the first half, 0-0 till the referee marked a non-hesistant penalty. Fortunately, the Nautico player missed the shot (penalty in soccer is regarded as a can't miss, something like a 30 yard field goal).
They returned to the second half playing a very close game, with no big opportunities to score, and the draw would promote Grêmio. However, at the 71', a Grêmio player was ejected, and the game got even thougher.
It was closing to a draw till the 82' minute, when the ref once again marked a non-hesistant penatly favoring the home team. The Grêmio players went nuts, and started charging towards the ref. The coach got on the field, it was a mess, and after 25 minutes of paralization, 3 Grêmio players were ejected. It was 11 players against 7, at the 103' minute (the game usually ends at 90, and this game would end at 115'), and the opposing team had yet another penalty to kick.
What happened was that Grêmio's goalkeeper saved the penalty. Nautico had missed his second penalty of the game.
There were still roughly 15 minutes to play, and they had a 4 man advantage.
After a lot of missed goal opportunities, Grêmio started a counterattack, that was paralized with a faul, and the Nautico player was ejected. It was 10x4 now.
But the Grêmio players were able to take Nautico by surprised. The faul was quickly taken, and Grêmio's last remaining offensive player was able to enter the adversary's area and score at the 108' minute.
The most improbable game of all time would end at 115'. The miracle had happened. Grêmio would emerge victorious and come back to Serie A, in a game that, for 30 minutes, they played with -1, and for 15 mintes, played with -4. A game where the opposing team had 2 opportunities to convert a penalty kick, yet failed.
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After all that, I re-read the thread and it says best
NFL game
... Well, I'd have to go with the Music City Miracle. Craziest, most exciting game I've ever seen.