  A (folklore?) conjecture states that no holomorphic modular form
$F(\tau)=\sum_{n=1}^\infty a_nq^n\in q\mathbb Z[[q]]$ exists, where $q=e^{2\pi
i\tau}$, such that its anti-derivative $\sum_{n=1}^\infty a_nq^n/n$ has
integral coefficients in the $q$-expansion. A recent observation of Broadhurst
and Zudilin, rigorously accomplished by Li and Neururer, led to examples of
meromorphic modular forms possessing the integrality property. In this note we
investigate the arithmetic phenomenon from a systematic perspective and discuss
related transcendental extensions of the differentially closed ring of
quasi-modular forms.
