This question just came to my mind when reading the question

When may Function (meromorphic) be expanded as power series with coefficients of integers

Suppose $f$ is an analytic function on some open subset $U \subseteq \mathbb{C}$. Are there sufficient or necessary conditions to be put on $f$ that $f$ has the following property. There exist a point $\zeta \in U$ such that the Taylor expansion of $f$ around $\zeta$ has only rational (algebraic) coefficients.

More generally, let us call the set of functions with this property $\mathcal{R}(U)$. It is easy to see that $\mathcal{R}(U)$ is dense in the set of all analytic functions with respect to the topology given by locally uniform convergence, since all polynomials with rational coefficients are contained in $\mathcal{R}(U)$ and are dense. But of course $\mathcal{R}(U)$ does not contain all analytic functions since for example $z^2+\pi$ or constant functions are not all contained. Also $\mathcal{R}(U)$ is uncountable, since $z-\zeta$ for any $\zeta \in \mathbb{C}$ satisfies the property.

So what can be said about $\mathcal{R}(U)$? Is it in some sense interesting?