The highest tile you can get is 2^17, 131072. Each doubling of the tile size requires another extra space, as you have to construct two of the previous tile, and once you construct one, it has to go somewhere. In the end, the final board will have a string of every power of two from 2^16 to 4, and then another 4 in the last empty space. The 4s then combine, then the 8s, and so on, rippling up like binary addition until you get 2^17. Hope that makes sense, I thought about this for a while but it's not the easiest concept to put to words.

If he got 32768, that would seem to be the highest unless you could get the following 16 squares which would collapse to 65536:

2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768

You could theoretically beat that if you happened to get a 4 just when you needed it (starting with 4, 4, 8, 16, etc.)