  We give conditions for the existence of the Minkowski content of limit sets
stemming from infinite conformal graph directed systems. As an application we
obtain Minkowski measurability of Apollonian gaskets, provide explicit formulae
of the Minkowski content, and prove the analytic dependence on the initial
circles. Further, we are able to link the fractal Euler characteristic, as well
as the Minkowski content, of Apollonian gaskets with the asymptotic behaviour
of the circle counting function studied by Kontorovich and Oh. These results
lead to a new interpretation and an alternative formula for the Apollonian
constant. We estimate a first lower bound for the Apollonian constant, namely
$0{.}055$, partially answering an open problem by Oh of 2013. In the higher
dimensional setting of collections of disjoint balls, generated e.\,g.\ by
Kleinian groups of Schottky type, we prove that all fractal curvature measures
exist and are constant multiples of each other. Further number theoretical
applications connected to the Gauss map and to the Riemann $\zeta$-function
illustrate our results.
