This involved hacks.  Ctrl-Shift-J and knowledge of Javascript made it easy to have the game start me with a bunch of well-placed tiles already on the board.  That said, if anyone wanted to see this game beaten, this is what it might look like.  Sorry that it takes a while for my phone to focus on the computer screen..

Edit November 2017 to address questions of if this is possible without hacks:

A computer processes at a few gigahertz.  That means that, even if everything else about this game was perfect and you had master ninja robot hands, you cannot get to 5 billion moves per second.

If you add up all the tiles on the board, before and after moving, the increase in that sum is equal to the tile it spawned.  The game spawns a 4 a tenth of the time, a 2 the rest of the time.  Thus on average, you add 2.2 points to the board each move.  You need to add 9 quadrillion points to the board before you win, so you need to move about 4 quadrillion times.

Making 4 quadrillion moves at the speed of the fastest computer…