  We show that for fixed $d>3$ and $n$ growing to infinity there are at least
$(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes
with $n$ vertices. This is about the square of the previous best lower bounds.
As an intermediate step, we show that certain neighborly polytopes (such as
particular realizations of cyclic polytopes) have at least $(n!)^{
\lfloor(d-1)/2\rfloor \pm o(1)}$ regular triangulations.
