The function $$ f(z)=\sum_{n=0}^{\infty} \frac{z^n}{2^n-1} $$ defines a holomorphic function for $|z|<2$, and it satisfies $$ f(2z) = f(z)+\frac{1}{1-z} $$ for $|z|<1$. Based on this identity, it is easy to prove that $f(z)$ extends to a meromorphic function on $\mathbb{C}$, and the set of poles is $\{2^n:\ n=1,2,\dots\}$. In particular, $f(z)$ does not define a rational function, because its meromorphic extension to $\mathbb{C}$ has infinitely many poles.
Regarding your second question, I recommend the work of Dwork (with which I am not familiar), e.g. (8) in Alain Robert's article "Des adeles: pourquoi", and Lemma 9 in Tao's blog.