Let $\mathbb{C}^*=\mathbb{C}-{0}$. What is the fundamental group $\mathbb{C}/G,$ where G is the group of homeomorphism $\{\phi^n ; n\in \mathbb{Z}\}$ with $\phi(z)=2z$?
I think the fundamental group of $\mathbb{C}^*$ is $\mathbb{Z}$. But I do not have an idea how to proceed next.
Hint:
The map $$\psi:\quad {\mathbb C}\to{\mathbb C}^*/G,\qquad w\mapsto e^w/_\sim$$ is a covering of your Riemann surface $R:={\mathbb C}^*/G$. Two points $w$, $w'$ map onto the same point of $R$ iff they are equivalent modulo the lattice generated by $\omega_1:=\log 2$, $\>\omega_2:=2\pi i$.