  We construct an explicit family of modular iterated integrals which involves
cusp forms. This leads to a new method of producing "invariant versions" of
iterated integrals of modular forms. The construction will be based on an
extension of higher-order modular forms which, in contrast to the standard
higher-order forms, applies to general Fuchsian groups of the first kind and,
as such, is of independent interest.
