  Holomorphic almost modular forms are holomorphic functions of the complex
upper half plane which can be approximated arbitrarily well (in a suitable
sense) by modular forms of congruence subgroups of large index in $\SL(2,\ZZ)$.
It is proved that such functions have a rotation-invariant limit distribution
when the argument approaches the real axis. An example for a holomorphic almost
modular form is the logarithm of $\prod_{n=1}^\infty (1-\exp(2\pi\i n^2 z))$.
The paper is motivated by the author's studies [J. Marklof, Int. Math. Res.
Not. {\bf 39} (2003) 2131-2151] on the connection between almost modular
functions and the distribution of the sequence $n^2x$ modulo one.
