The cycle pairing on graphs takes a pair of cycles to their oriented intersection. While purely combinatorial, it arises in Picard-Lefschetz theory as a way of studying monodromy of families of algebraic curves, variations of Hodge structures, and asymptotics of period integrals. The cycle pairing, once properly packaged, determines a graph up to two moves by the graph Torelli theorem of Caporaso and Viviani. In this talk, we discuss tropical iterated integrals, a mildly non-Abelian extension of the cycle pairing. We relate them to asymptotics of iterated integrals and monodromy on the fundamental group. We discuss the obstructions to a more precise unipotent Torelli theorem. This is joint work with Raymond Cheng.