  Hurwitz numbers count genus $g$, degree $d$ covers of the complex projective
line with fixed branched locus and fixed ramification data. An equivalent
description is given by factorisations in the symmetric group. Simple double
Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In
recent years a related counting problem in the context of random matrix theory
was introduced as so-called monotone Hurwitz numbers. These can be viewed as a
desymmetrised version of the Hurwitz-problem. A combinatorial interpolation
between simple and monotone double Hurwitz numbers was introduced as mixed
double Hurwitz numbers and it was proved that these objects are piecewise
polynomial in a certain sense. Moreover, the notion of strictly monotone
Hurwitz numbers has risen interest as it is equivalent to a certain
Grothendieck dessins d'enfant count. In this paper, we introduce a
combinatorial interpolation between simple, monotone and strictly monotone
double Hurwitz numbers as \textit{triply interpolated Hurwitz numbers}. Our aim
is twofold: Using a connection between triply interpolated Hurwitz numbers and
tropical covers in terms of so-called monodromy graphs, we give algorithms to
compute the polynomials for triply interpolated Hurwitz numbers in all genera
using Erhart theory. We further use this approach to study the wall-crossing
behaviour of triply interpolated Hurwitz numbers in genus $0$ in terms of
related Hurwitz-type counts. All those results specialise to the extremal cases
of simple, monotone and Grothendieck dessins d'enfants Hurwitz numbers.
