  Given a klt singularity $(X,\Delta;x)$, we define the iteration of Cox rings
of $(X,\Delta;x)$. The first result of this article is that the iteration of
Cox rings ${\rm Cox}^{(k)}(X,\Delta;x)$ of a klt singularity stabilizes for $k$
large enough. The second result is a boundedness one, we prove that for a
$n$-dimensional klt singularity $(X,\Delta;x)$ the iteration of Cox rings
stabilizes for $k\geq c(n)$, where $c(n)$ only depends on $n$. Then, we use Cox
rings to establish the existence of a simply connected factorial canonical
cover (or scfc cover) of a klt singularity. We prove that the scfc cover
dominates any sequence of quasi-\'etale finite covers and reductive abelian
quasi-torsors of the singularity. We characterize when the iteration of Cox
rings is smooth and when the scfc cover is smooth. We also characterize when
the spectrum of the iteration coincides with the scfc cover. Finally, we give a
complete description of the regional fundamental group, the iteration of Cox
rings, and the scfc cover of klt singularities of complexity one. Analogous
versions of all our theorems are also proved for Fano type morphisms. To extend
the results to this setting, we show that the Jordan property holds for the
regional fundamental group of Fano type morphisms.
