  Given an abstract $n$-polytope $\mathcal{K}$, an abstract $(n+1)$-polytope
$\mathcal{P}$ is an extension of $\mathcal{K}$ if all the facets of
$\mathcal{P}$ are isomorphic to $\mathcal{K}$. A chiral polytope is a polytope
with maximal rotational symmetry that does not admit any reflections. If
$\mathcal{P}$ is a chiral extension of $\mathcal{K}$, then all but the last
entry of the Schl\"afli symbol of $\mathcal{P}$ are determined. In this paper
we introduce some constructions of chiral extensions $\mathcal{P}$ of certain
chiral polytopes in such a way that the last entry of the Schl\"afli symbol of
$\mathcal{P}$ is arbitrarily large.
