We will discuss certain sequences (R_0, R_1, ...) of symmetric functions that we call sprout symmetric functions. Many examples of sprout symmetric functions have already appeared in the literature, connected with such topics as the symmetric function generalization of the Tutte polynomial of a graph, Hirzebruch's L-genus, and zeta polynomials of binomial posets. We first consider basic properties of sprout symmetric functions, in particular, their expansion into classical bases for symmetric functions. Especially interesting is the Schur function expansion, which is closely related to the Edrei-Thoma theorem from the theory of total positivity. We conclude with some examples that unify and generalize results from the theory of permutation enumeration.