This is indeed very well-known. There are plenty of counterexamples. In fact, there's a remarkable construction due to van der Poorten of an a "transcendental" entire function $f$ such that $f$ and all its derivatives map any algebraic number $\alpha$ into $\mathbb Q (\alpha)$. See
A.J. van der Poorten, Transcendental entire functions mapping every algebraic number field into itself, J. Austral.Math. Soc. 8 (1968), 192–193.