Using your Head is Permitted

The game is played on a 4x4 grid. In the beginning of each turn, one of the grid's empty positions is randomly chosen to become a square with either a "2" or a "4" printed on it, where the choice of value is also random. (In the beginning of the first turn, two such grid positions are chosen and the rest are empty.) The player must then choose one of four directions (up, down, left, right) and consequently the squares currently on the board "fall" in the chosen direction. If the fall causes two squares with identical numbers to collide, they become a single square whose number is the sum of the original two numbers. (The exact rules governing these collisions can be discovered by playing the game. Here is the official site. Caution: may be habit-forming.) Note that choosing a direction that causes no movement of the squares on the grid, whether by fall or by collision, is not a legal move.

Winning the game involves reaching a point where one of the squares on the grid has the number "2048". However, even after this goal is reached the game can be played on. It terminates only when the entire grid is filled and there is no empty grid point in which a square can be added to start the next turn.

This month's Using your Head is Permitted riddle relates to a slight variation over "2048", chosen to make analysis easier. In this variant, if a player has created a collision on the previous move and if, additionally, the position reached is one in which a further move can cause another collision (e.g., if two squares with identical values are adjacent), then no new square will appear on the grid at the start of the next round. In all other respects, this is the standard "2048" game.

This month's riddle: determine, with proof, what is the largest value that can ever appear in any square during the game. (I would assume that the game's designer, Gabriele Cirulli, had to calculate this value, too, in order to make sure that there is enough space on each of the game's grid squares in order to print it.)

List of solvers: