I love Europa Universalis 4 (PC game), the game starts on November 11th, 1444 - a day after the Hungarian/Christian defeat in the battle of Varna, which paved the way for the Ottomans becoming a superpower.

That's a cool date for the game, because it allows the players to try to save a nearly doomed Byzantine Empire, or change the outcome of the 100 Year War for example.