If your definition of holomorphic is that there is an absolutely convergent power series $\sum_{n=0}^{\infty} a_n z^n$ then Parseval's theorem implies that the mean square of the function on the circle of radius $R$ is $\sum_{n=0}^{\infty} |a_n^2| R^{2n}$ and it follows that $a_n=0$ for all $n>0$.
Edit (new approach)
Without loss of generality let's assume that $f(0)=0$ and $f'(0)=0$. Consider the function $f(x)/x$, continuous on all of $\mathbb{C}$. It is analytic on the punctured plane and converges to 0 on its boundary, hence by the maximum principle must be 0.
You can drop the $f'(0)=0$ condition above if you assume instead that $f$ is twice differentiable at 0, so then $f(x)/x$ is holomorphic on all of $\mathbb{C}$ and decreases to 0, so must be identically 0.
This does presuppose the maximum principle.
