It is worth mentioning that in the original game you can choose to keep going after you reach 2048. This game had to remove that option to achieve a 64bit state. Still, a clever implementation.
Since 15^16 < 2^64, you can still use a 64 bit state to reach 2^15=32768, which does not seem to be reachable in practice (the previous state would fill the whole board!)
edit: Explanation of the 131072 cap: https://puzzling.stackexchange.com/questions/48/what-is-the-...
Also, in one of the answers there someone claims to have achieved 65536 in practice
This is a partial video of one of them: https://youtube.com/watch?v=QQSLjPHg5P8
The state in this implementation also stores a random seed (between 0 and 99, exclusive), so using 16^16 for the state would leave nothing for the random seed.