I'm not going to write it out, there is certainly a proof that the list is infinite in base 2^k (for integer k >= 2). I'm more wondering about how hard it is to prove that the list is finite in a different base.
if we marked sequences of integers with 3 options. even, odd, other. then these lists are not finite in bases of 3^k.
for four options. even, odd, other, another. then these lists are not finite in bases of 4^k.
there is an intersection in the infinite lists where the base is equivalent to the power of an earlier base.
so infinite lists for 2^k would overlap a subset of the infinite lists for 2^2^k=4^k
all prime bases, p, p^k would admit infinite lists that cover all the infinite lists for some composite base, c, c^k.
similarly there the largest number with all prime digits actually differs if you ask the question in different bases.
and there is also a pattern that exists to predict what the number will be in a given base.
do you mean the largest prime number where all digits are prime numbers?