Two relevant references:
- W. Abikoff, Some remarks on Kleinian groups. 1971 Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) pp. 1–5. Ann. of Math. Studies, No. 66. Princeton Univ. Press, Princeton, N.J.
Among other things, he constructs an infinitely generated free Kleinian subgroup $\Gamma< PSL(2,C)$ whose limit set is a Jordan curve of positive planar measure.
It is worth looking more closely at his construction to see if it can be made using fundamental domains with a single accumulation point of boundary faces.
- K. Matsuzaki, The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups. Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 1, 123–139.
In Theorem 2' he proves that if $\Gamma< PSL(2,R)$ is a discrete subgroup such that the convex core $C$ of $H^2/\Gamma$ consists of infinitely many boundary loops $c_i$ which satisfy $$ \sum_{i} \ell(c_i)^{1/2}<\infty $$ then the limit set of $\Gamma$ (in $S^1$) has positive linear measure. (Here $\ell$ denotes the length of a curve) Using this it is easy to construct an example of an infinite rank Schottky subgroup of $PSL(2,R)$ whose fundamental domain has unique accumulation point (of pairwise disjoint boundary arcs) and such that the limit set has positive linear measure. This is a 1-dimensional version of an example you are asking for.
Edit. There is a variety of inequivalent definitions of Schottky groups in the literature. An easy and well-known construction (which some people call "Schottky" but you do not!) is the following:
Start with a compact nowhere dense subset $K\subset R^2$ of positive measure (say the product of two thick Cantor sets in $R$). Let $D_i\subset R^2$ be a collection of pairwise disjoint closed disks disjoint from $K$, such that the closure of $$ \bigcup_{i} D_i $$ contains $K$. Now, take the group $\Gamma_K< Mob(S^2)$ generated by inversions in the boundary circles of the disks $D_i$. Its limit set will contain $K$ and, hence, will have positive measure. By taking $K$ totally disconnected one obtains examples with totally disconnected limit sets.
I think this is what Rich Schwartz had/has in mind.
This is all fine and well, but you want the union of disks to accumulate at a single point in $S^2$. Here is the essential difference between the two classes of groups: The above example $\Gamma_K$ will have dissipative action on the limit set (i.e. there is a measurable wondering domain of positive measure, namely, $K$). On the other hand, you are asking for a Schottky group such that the action on the limit set is conservative, i.e. admits no measurable wondering domains (this is not immediate, but follows from Sullivan's work; the key thing is that the closure of the $H^3$-fundamental domain in your case intersects the limit set in a subset of measure zero, since it is a singleton).