  We study the growth of double cosets in the class of groups with contracting
elements, including relatively hyperbolic groups, CAT(0) groups and mapping
class groups among others. Generalizing a recent work of Gitik and Rips about
hyperbolic groups, we prove that the double coset growth of two Morse subgroups
of infinite index is comparable with the orbital growth function. The same
result is further obtained for a more general class of subgroups whose limit
sets are proper subsets in the entire limit set of the ambient group.
  As an application, we confirm a conjecture of Maher that hyperbolic
3-manifolds are exponentially generic in the set of 3-manifolds built from
Heegaard splitting using complexity in Teichm\"{u}ller metric.
