Triple Equivariant Eisenstein Integrals

We present a theorem for producing length three modular iterated integrals which are not simply linear combinations of real analytic Eisenstein series; this can be viewed as an extension of the length two case recently given by F. Brown. These modular iterated integrals sit inside the space of real analytic modular forms and, conjecturally, they can be used to give explicit expressions of important modular graph functions.