A conjecture of Colin and Honda states that the number of periodic Reeb orbits of
universally tight contact structures on hyperbolic manifolds grows exponentially with
the period, and they speculate further that the growth rate of contact homology is
polynomial on nonhyperbolic geometries. Along the line of the conjecture, for
manifolds with a hyperbolic component that fibers on the circle, we prove
that there are infinitely many nonisomorphic contact structures for which
the number of periodic Reeb orbits of any nondegenerate Reeb vector field
grows exponentially. Our result hinges on the exponential growth of contact
homology, which we derive as well. We also compute contact homology in
some nonhyperbolic cases that exhibit polynomial growth, namely those of
universally tight contact structures on a circle bundle nontransverse to the
fibers.