I would not argue with the irrationality of the function following from the functional equation relating $f(2z)$ to $f(z)$. In fact this function is a particular case of the so-called $q$-logarithm which received a quite special attention. Note that being rational over $\mathbb C(z)$ or over $\mathbb Q(z)$ for a power series with rational coefficients is equivalent.

If $f(z)$ where rational then $$f(1)=\sum_{n=1}^\infty \frac1{2^n-1}=\sum_{m=1}^\infty \tau(m)2^{-m}$$ would be a rational number, where $\tau(m)$ denotes the number of divisors of $m$. Erd\H{o}s proved already in 1948 that this is not the case by showing that the base $2$ expansion of the number is not periodic. (This is still a nice exercise in analytic number theory!)

The irrationality of the values of $f(z)$ for rational $z\ne0$ was established by P. Borwein in 1992 using the Pad'e approximations to the function.

I would not argue with the irrationality of the function following from the functional equation relating $f(2z)$ to $f(z)$. In fact this function is a particular case of the so-called $q$-logarithm which received a quite special attention. Note that being rational over $\mathbb C(z)$ or over $\mathbb Q(z)$ for a power series with rational coefficients is equivalent.

If $f(z)$ where rational then $$f(1)=\sum_{n=1}^\infty \frac1{2^n-1}=\sum_{m=1}^\infty \tau(m)2^{-m}$$ would be a rational number, where $\tau(m)$ denotes the number of divisors of $m$. Erdős proved already in 1948 that this is not the case by showing that the base $2$ expansion of the number is not periodic. (This is still a nice exercise in analytic number theory!)

The irrationality of the values of $f(z)$ for rational $z\ne0$ was established by P. Borwein in 1992 using the Padé approximations to the function.