The total of all the numbers on the board goes up by 2 on every move and starts at 4. Therefore the number of moves you've made is (current total - 4)/2. To reach 2048 requires at least 1022 moves. As a rough rule of thumb, look at the biggest number on the board: that's a ballpark estimate of how many moves you've played.

I realised this shortly after getting into the flow of my first game, and it entirely extinguished my desire to play a second. My memetic immune system is strong.