We develop a class of integrals on a manifold\ *M*\ called\ *exponential iterated integrals \ *, an extension of K.T. Chen\&$\#$39;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.