  Elliptic modular graph forms (eMGFs) are non-holomorphic modular forms
depending on a modular parameter $\tau$ of a torus and marked points $z$
thereon. Traditionally, eMGFs are constructed from nested lattice sums over the
discrete momenta on the worldsheet torus in closed-string genus-one amplitudes.
In this work, we develop methods to translate the lattice-sum realization of
eMGFs into iterated integrals over modular parameters $\tau$ of the torus with
particular focus on cases with one marked point. Such iterated-integral
representations manifest algebraic and differential relations among eMGFs and
their degeneration limit $\tau \rightarrow i\infty$. From a mathematical point
of view, our results yield concrete realizations of single-valued elliptic
polylogarithms at arbitrary depth in terms of meromorphic iterated integrals
over modular forms and their complex conjugates. The basis dimensions of eMGFs
at fixed modular and transcendental weights are derived from a simple counting
of iterated integrals and a generalization of Tsunogai's derivation algebra.
