Let $n$ be a positive integer. Given an integer base $b\ge 2$, let $C_b(n)$ be the number of *non-zero* digits in the expansion of $N$ in base $b$. Further, let $M(n)=\max\{C_b(n):b\ge 2\}$ be the maximum of all $C_b$. Obviously, for each $n$ only $b\le n$ count, and $M(n)\le 1+\log_2 n$.

I would like to understand the relation between the $C_b(n)$ for different bases $b$. It seems reasonable to conjecture (and I'm sure stronger conjectures exist) that if $b, b'$ are multiplicatively independent, then $C_b(n)+C_{b'}(n)>\varepsilon \log n$ for some $\varepsilon(b,b')>0$ and all $n$. However, this seems to be out of reach. The best I have found are estimates of the form $C_b(n)+C_{b'}(n)\ge \frac{\log\log n}{\log\log\log n}$, using Baker's Theorem.

My question is:

Can we do better by considering more bases? In particular, is it true that there is $\varepsilon>0$ such that $M(n) \ge \varepsilon \log n$ for all $n$?

The special case in which $n$ is a power would already be interesting, and is perhaps a little easier:

Does there exist $\varepsilon>0$ such that $M(2^k) > \varepsilon k$ for all $k$?

This question came up in studying the absolute continuity of the convolution of certain self-similar measures, a question at the interface of fractal geometry/dynamics/number theory.