The various entropy bounds that exist in the literature suggest that spacetime is fundamentally discrete, and hint at an underlying relationship between geometry and ``information''. The foundation of this relationship is yet to be uncovered, but should manifest itself in a theory of quantum gravity. We present a measure for the maximal entropy of spherically symmetric spacelike regions within the causal set approach to quantum gravity. In terms of the proposal, a bound for the entropy contained in this region can be derived from a counting of potential ``degrees of freedom'' associated to the Cauchy horizon of its future domain of dependence. For different spherically symmetric spacelike regions in Minkowski spacetime of arbitrary dimension, we show that this proposal leads, in the continuum approximation, to Susskind's well-known spherical entropy bound.