  We derive new Poincar\'e-series representations for infinite families of
non-holomorphic modular invariant functions that include modular graph forms as
they appear in the low-energy expansion of closed-string scattering amplitudes
at genus one. The Poincar\'e series are constructed from iterated integrals
over single holomorphic Eisenstein series and their complex conjugates,
decorated by suitable combinations of zeta values. We evaluate the Poincar\'e
sums over these iterated Eisenstein integrals of depth one and deduce new
representations for all modular graph forms built from iterated Eisenstein
integrals at depth two. In a companion paper, some of the Poincar\'e sums over
depth-one integrals going beyond modular graph forms will be described in terms
of iterated integrals over holomorphic cusp forms and their L-values.
