  We study generating series of torus integrals that contain all so-called
modular graph forms relevant for massless one-loop closed-string amplitudes. By
analysing the differential equation of the generating series we construct a
solution for its low-energy expansion to all orders in the inverse string
tension $\alpha'$. Our solution is expressed through initial data involving
multiple zeta values and certain real-analytic functions of the modular
parameter of the torus. These functions are built from real and imaginary parts
of holomorphic iterated Eisenstein integrals and should be closely related to
Brown's recent construction of real-analytic modular forms. We study the
properties of our real-analytic objects in detail and give explicit examples to
a fixed order in the $\alpha'$-expansion. In particular, our solution allows
for a counting of linearly independent modular graph forms at a given weight,
confirming previous partial results and giving predictions for higher, hitherto
unexplored weights. It also sheds new light on the topic of uniform
transcendentality of the $\alpha'$-expansion.
