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A generic prediction of string theory is the existence of many axion fields. It has recently been argued that many of these fields should be light and, like the well-known QCD axion, lead to observable cosmological consequences. In this paper, we study in detail the effect of the so-called string axiverse on large-scale structure, focusing on the morphology and evolution of density perturbations, anisotropies in the cosmic microwave background, and weak gravitational lensing of distant galaxies. We quantify specific effects that will arise from the presence of the axionic fields and highlight possible degeneracies that may arise in the presence of massive neutrinos. We take particular care understanding the different physical effects and scales that come into play. We then forecast how the string axiverse may be constrained and show that with a combination of different observations, it should be possible to detect a fraction of ultralight axions to dark matter of a few percent. [ABSTRACT FROM AUTHOR]