In the game 2048, what is the biggest theoretical tile?

 

In the game 2048 what is the biggest tile that can be achieved, assuming a player playing optimally and tile spawning at the most optimal place?

Naively I would say that the biggest achievable tile is 65536 * 2 = 131072 because it seems that the best possible board is the following: 

  4     4     8     16
 256   128    64    32
 512   1024  2048  4096
65536 32768 16384  8192

But I'm not sure if

  • it's correct
  • how to prove that my intuition is indeed correct.

    • (sorry if I should have asked on gaming.stackexchange, but this is more of a CS question than a gaming one afaict)

    you haven't finished yet with the board you propose: you can slide to the right, going all the way down and obtaining 131072. So your analysis was correct although you missed a spot:

    This will be your final board: 

      4     8     16    32
 512   256   128    64
 1024  2048  4096  8192
131072 65536 32768 16384
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