Yes. This is due Z.-X. He, and in greater generality to Rich Schwartz, see
Schwartz, Richard, The limit sets of some infinitely generated Schottky groups, Trans. Am. Math. Soc. 335, No.2, 865-875 (1993). ZBL0815.30033.
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.