The digital sum of a number is the sum of its digits, and it's similarly defined for digital products.
Let's define the digital exponentiation of a number to be the evaluation of the exponentiation of all of the digits written in a tower. So for example:
$f(345) = {3^4}^5 = (3^4)^5=81^5=3486784401$
For some values, like 200, this will obviously converge to a single-digit number, and the digital exponentiation of a single digit is just that digit. For others, it appears that they will diverge infinitely.
However, is there a way to prove that it will always either converge to a single digit or diverge infinitely? As in, does it ever stabilize at some other value(s), or form a cycle?