Yes. This is due Z.-X. He, and in greater generality to Rich Schwartz, see

<cite authors="Schwartz, Richard">_Schwartz, Richard_, [**The limit sets of some infinitely generated Schottky groups**](http://dx.doi.org/10.2307/2154409), Trans. Am. Math. Soc. 335, No.2, 865-875 (1993). [ZBL0815.30033](https://zbmath.org/?q=an:0815.30033).</cite>

For a packing with positive area, the following is given by this same Rich Schwartz (private communication):

Suppose you look at the Schottky group generated by inversions
in disks of the Apollonian gasket.  Call this union of disks $D_0.$
Note that $D_0$ has full measure in the sphere.  Let $D_1$ be the unions,
taken over all disks of $D_0,$ of the images of $D_0$ under inversion
in the disks of $D_0.$  Then $D_1$ also has full measure in the sphere and
$D_1 \subset D_0.$  Next define $D_2 \subset D_1,$ etc.  The nested
intersection $D_0 \cap D_1 \cap D_2\dots$  again has full measure in the
sphere and I think that every point in this intersection is a
point of the limit set.