We study the problem of classifying Kleinian groups via the topology of their limit sets. In particular, we are interested in one-ended convex-cocompact Kleinian groups where each piece in the JSJ decomposition is a free group, and we describe interesting examples in this situation. In certain cases we show that the type of Kleinian group is determined by the topology of its group boundary. We conjecture that this is not the case in general. We also determine the homeomorphism types of planar boundaries that can occur. This is joint work in progress with Peter Haissinsky and Luisa Paoluzzi.