Abstract: In this talk, I will characterize the late-time expansion rate of the universe in scalar-field cosmologies with multi-exponential potentials. To do this, I will take advantage of previously unobserved universal asymptotic features of the solutions to the cosmological equations. This is a critical achievement because it provides a simple diagnostic of whether any given multi-exponential potential holds the necessary conditions for late-time cosmic acceleration. Such potentials have been studied extensively as phenomenological models of quintessence and, moreover, they are ubiquitous in string-theoretic constructions, which further allows one to sharpen several statements on the low-energy signatures of quantum gravity. I will extensively comment on the tension for acceleration posed by the string-theoretic dilaton, which, if present as a rolling scalar in a weakly-coupled theory, makes the potential too steep for acceleration unless negative-definite potentials are present. I will also show that, if the late-time scale factor is of power-law form, the acceleration parameter can be expressed in terms of a directional derivative of the potential.