  We study Calabi-Yau threefolds with large Hodge numbers by constructing and
counting triangulations of reflexive polytopes. By counting points in the
associated secondary polytopes, we show that the number of fine, regular, star
triangulations of polytopes in the Kreuzer-Skarke list is bounded above by
$\binom{14,111}{494} \approx 10^{928}$. Adapting a result of Anclin on
triangulations of lattice polygons, we obtain a bound on the number of
triangulations of each 2-face of each polytope in the list. In this way we
prove that the number of topologically inequivalent Calabi-Yau hypersurfaces
arising from the Kreuzer-Skarke list is bounded above by $10^{428}$. We
introduce efficient algorithms for constructing representative ensembles of
Calabi-Yau hypersurfaces, including the extremal case $h^{1,1}=491$, and we
study the distributions of topological and physical data therein. Finally, we
demonstrate that neural networks can accurately predict these data once the
triangulation is encoded in terms of the secondary polytope.
