Siegel introduced the notion of E-function in a landmark 1929 paper with the goal of generalising the Hermite-Lindemann-Weierstrass theorem on the transcendence of the values of the exponential function at algebraic numbers. E-functions are power series with algebraic coefficients that are solutions of a linear differential equation and satisfy some growth conditions of arithmetic nature. Besides the exponential function, examples include Bessel functions and a rich family of hypergeometric series. Siegel asked whether all E-functions are polynomial expressions in these hypergeometric series. I explain why the answer to Siegel's question is negative, and then try to amend it by describing how E-functions arise from geometry in the form of "exponential period functions" and why it might seem reasonable, in the light of other conjectures, to expect that all E-functions are of this kind.