We develop a class of integrals on a manifold *M* called *exponential iterated integrals *, an extension of K.T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of π_{1}(M,x) can be expressed via these new integrals. The ring of exponential iterated integrals contains the coordinate rings for a class of universal representations, called the *relative solvable completions * of π_{1}(M,x). We consider exponential iterated integrals in the particular case of fibered knot complements, where the fundamental group always has a faithful relative solvable completion.