I wonder if we can get a sense of how fast it would grow if we hypothesize it is an infinite sequence. And if it is a finite sequence, one could define f(p, n)

I wonder if we can get a sense of how fast it would grow if we hypothesize it is an infinite sequence. And if it is a finite sequence, one could define f(p, n) as the sequence of successive exponents of 2 such that the ratio of even digits over its total number of digits is greater than p. This could be an interesting way of describing a set of fast growing functions from exponential growth (p=0) to arbitrarily fast growth as p grows closer to 1 (or P where P is the smallest number such that f(P, n) is a finite sequence).