  Given a non-empty bounded subset of hyperbolic space and a Kleinian group
acting on that space, the orbital set is the orbit of the given set under the
action of the group. We may view orbital sets as bounded (often fractal)
subsets of Euclidean space. We prove that the upper box dimension of an orbital
set is given by the maximum of three quantities: the upper box dimension of the
given set; the Poincar\'e exponent of the Kleinian group; and the upper box
dimension of the limit set of the Kleinian group. Since we do not make any
assumptions about the Kleinian group, none of the terms in the maximum can be
removed in general. We show by constructing an explicit example that the
(hyperbolic) boundedness assumption on $C$ cannot be removed in general.
