[en] In quantum superdense coding, two parties previously sharing entanglement can communicate a two bit message by sending a single qubit. We study this feature in the broader framework of general probabilistic theories. We consider a particular class of theories in which the local state space of the communicating parties corresponds to Euclidian hyperballs of dimension n (the case n = 3 corresponds to the Bloch ball of quantum theory). We show that a single n-ball can encode at most one bit of information, independently of n. We introduce a bipartite extension of such theories for which there exist dense coding protocols such that ${\mathrm{l}\mathrm{o}\mathrm{g}}_2(n+1)$ bits are communicated if entanglement is previously shared by the communicating parties. For $n><!--\; >\; -->3,$ these protocols are more powerful than the quantum one, because more than two bits are communicated by transmission of a system that locally encodes at most one bit. We call this phenomenon hyperdense coding (HDC). Our HDC protocols imply superadditive classical capacities: two entangled systems can encode ${\mathrm{l}\mathrm{o}\mathrm{g}}_2(n+1)><!--\; >\; -->2$ bits, even though each system individually encodes at most one bit. In our examples, HDC and superadditivity of classical capacities come at the expense of violating tomographic locality or dynamical continuous reversibility. (paper)