  We study the well-posedness of triply nonlinear degenerate
elliptic-parabolic-hyperbolic problem $$ b(u)_t - {\rm div} \tilde{\mathfrak
a}(u,\nabla\phi(u))+\psi(u)=f, \quad u|_{t=0}=u_0 $$ in a bounded domain with
homogeneous Dirichlet boundary conditions. The nonlinearities $b,\phi$ and
$\psi$ are supposed to be continuous non-decreasing, and the nonlinearity
$\tilde{\mathfrak a}$ falls within the Leray-Lions framework. Some restrictions
are imposed on the dependence of $\tilde{\mathfrak a}(u,\nabla\phi(u))$ on $u$
and also on the set where $\phi$ degenerates. A model case is $\tilde{\mathfrak
a}(u,\nabla\phi(u))
=\tilde{\mathfrak{f}}(b(u),\psi(u),\phi(u))+k(u)\mathfrak{a}_0(\nabla\phi(u)),$
with $\phi$ which is strictly increasing except on a locally finite number of
segments, and $\mathfrak{a}_0$ which is of the Leray-Lions kind. We are
interested in existence, uniqueness and stability of entropy solutions. If
$b=\mathrm{Id}$, we obtain a general continuous dependence result on data
$u_0,f$ and nonlinearities $b,\psi,\phi,\tilde{\mathfrak{a}}$. Similar result
is shown for the degenerate elliptic problem which corresponds to the case of
$b\equiv 0$ and general non-decreasing surjective $\psi$. Existence, uniqueness
and continuous dependence on data $u_0,f$ are shown when $[b+\psi](\R)=\R$ and
$\phi\circ [b+\psi]^{-1}$ is continuous.
