  We prove that the number of critical points of a Li-Tam Green's function on a
complete open Riemannian surface of finite type admits a topological upper
bound, given by the first Betti number of the surface. In higher dimensions, we
show that there are no topological upper bounds on the number of critical
points by constructing, for each nonnegative integer $N$, a manifold
diffeomorphic to $\RR^n$ ($n \geq 3$) whose minimal Green's function has at
least $N$ nondegenerate critical points. Variations on the method of proof of
the latter result yield contractible $n$-manifolds whose minimal Green's
functions have level sets diffeomorphic to any fixed codimension $1$ compact
submanifold of $\RR^n$.
