  We discuss the stabilization of symmetric products Sym^n(X) of a smooth
projective variety X in the Grothendieck ring of varieties. For smooth
projective surfaces X with non-zero h^0(X, \omega_X), these products do not
stabilize; we conditionally show that they do not stabilize in another related
sense, in response to a question of R. Vakil and M. Wood. There are analogies
between such stabilization, the Dold-Thom theorem, and the analytic class
number formula. Finally, we discuss Hodge-theoretic obstructions to the
stabilization of symmetric products, and provide evidence for these
obstructions in terms of a relationship between the Newton polygon of a certain
"motivic zeta function" associated to a curve, and its Hodge polygon.
