  A space of modular iterated integrals sits inside the space of real analytic
modular forms. We present a theorem for producing length three modular iterated
integrals which are not simply combinations of real analytic Eisenstein series;
each function has an associated Laplace-eigenvalue equation. This can be viewed
as an extension of the length two case recently given by F. Brown, a review of
which is included in this paper. We discuss how modular iterated integrals
could help understand the modular graph functions which arise in string
perturbation theory.
