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

Laplace-eigenvalue equations for length three modular iterated integrals