  We systematically analyze the fibration structure of toric hypersurface
Calabi-Yau threefolds with large and small Hodge numbers. We show that there
are only four such Calabi-Yau threefolds with $h^{1, 1} \geq 140$ or $h^{2, 1}
\geq 140$ that do not have manifest elliptic or genus one fibers arising from a
fibration of the associated 4D polytope. There is a genus one fibration
whenever either Hodge number is 150 or greater, and an elliptic fibration when
either Hodge number is 228 or greater. We find that for small $h^{1, 1}$ the
fraction of polytopes in the KS database that do not have a genus one or
elliptic fibration drops exponentially as $h^{1,1}$ increases. We also consider
the different toric fiber types that arise in the polytopes of elliptic
Calabi-Yau threefolds.
