Stack the digits of a natural number into a power tower, iterate until only one digit remains.
Does this iteration always terminate for any positive integer?
Additionally specify $0^n = 0$, even when $n=0$.
For example
$$ \begin{aligned} f_0&=58\\ f_1&=5^8=390625\\ f_2&=3^{9^{0}}=3^1=3\\ \end{aligned} $$
The smallest uncalculable number is 28:
$$ \begin{aligned} 28 &\to 256 \to 39235776294252497421590\cdots\\ 44 &\to \color{red}{256} \\ 48 &\to 65536\\ 49 &\to 262144\to6871\cdots\to28251\cdots\\ 54 &\to 625 \to 7958661\cdots\\ 57 &\to 78125 \to 5764801\\ 65 &\to 7776 \\ 66 &\to 46656 \\ 67 &\to 279936\\ 72 &\to \color{red}{49}\\ 73 &\to 343 \to 3433683820\cdots\\ 77 &\to 823543\\ 78 &\to 5764801\\ 85 &\to 32768\\ 86 &\to 262144\\ 97 &\to 4782969\\ 99 &\to 387420489\\ \end{aligned} $$
For number $5243$, we have $5^{2^{4^3}}=5243909\dots$ which cannot terminate.