An elementary example is given by the formula 
$$f(x)=\sum_{k=1}^\infty k!\,\Phi(x-k!)$$
for real $x>1$, where $\Phi$ is the standard normal cumulative distribution function. Then $f$ is real analytic on $(1,\infty)$ and $f(x)=O(x)$ for real $x>1$, whereas for natural $m$ we have 
$$\frac{f(m!)}{f(m!/2)}\sim\frac{m!/2}{(m-1)!}=\frac m2\to\infty$$
as $m\to\infty$.