Suppose that $f$ is analytic on an open set $D\subset\mathbb{C}$, and one uses Runge's theorem to obtain a sequence of rational functions $\{r_n\}$ which approach $f$ uniformly on compact subsets of $D$. If $f$ is not a rational function then $\deg(r_n)\to\infty$ as $n\to\infty$.
See answer below for the proof.
Suppose by way of contradiction that $\deg(r_n)\not\to\infty$. Then by dropping to a subsequence we may assume without loss of generality that $\deg(r_n)$ is a constant $k$. Since each component of $D^c$ is compact in the Riemann sphere, one may again drop to a subsequence and obtain that the zeros and poles of $r_n$ approach a set of $k$ points (counted with multiplicity). Therefore $r_n$ converges to a rational function $r$ of degree less than or equal to $k$ (possibly less than if a zero and a pole of $r_n$ are approaching the same point). This implies that $f=r$, a rational function.