I was always wondering the following:
Given a real analytic function there exists a positive radius of convergence for every point. This won't be affected by allowing complex numbers so it extends around every point to the complex plane within some disk.
Now I ask myself is there necessarily smallest strictly positive radius of convergence for every point simultaneously yielding a whole stripe around the real axis?
There seems to be strong evidence that this will fail in general as a countable number of poles might appear arbitrarily close to the real axis but I'm not sure as constructing such a function would same time involve a limiting process by adding countably many poles to it...
No, it is possible that the radii of convergence become arbitrarily small.
We can write down explicit examples, e.g.
$$f(z) = \frac{1}{\sin \left(\pi(z^2-i)\right)}$$
or
$$g(z) = \frac{1}{\sin \left(\pi(z^2-i)\right)} + \frac{1}{\sin \left(\pi (z^2+i)\right)}$$
if you prefer functions that are real-valued on $\mathbb{R}$.
More generally, every domain in the complex plane is the domain of existence of some holomorphic functions (that means, the function cannot be analytically continued across any part of the boundary), so for every (connected) open subset $U$ of $\mathbb{C}$ containing $\mathbb{R}$ there are real-analytic functions on $\mathbb{R}$ such that the radius of convergence of the Taylor series about $x$ is $\operatorname{dist}(x,\partial U)$.