  The work of J{\o}rgensen and Thurston shows that there is a finite number
N(v) of orientable hyperbolic 3-manifolds with any given volume v. In this
paper, we construct examples showing that the number of hyperbolic knot
complements with a given volume v can grow at least factorially fast with v. A
similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn
surgery. Furthermore, we give explicit estimates for lower bounds of N(v) in
terms of v for these examples. These results improve upon the work of Hodgson
and Masai, which describes examples that grow exponentially fast with v. Our
constructions rely on performing volume preserving mutations along Conway
spheres and on the classification of Montesinos knots.
