This question is motivated by a MathOverflow question by Peng Zhang. Valiant showed that counting maximal cliques in a general graph is #P-complete, but what if we restrict to incomparability graphs (i.e., we want to count maximal antichains in a finite poset)? This question seems natural enough that I suspect that it has been considered before, but I have not been able to locate it in the literature.
According to this abstract for "The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected" (SIAM J. Comput. 12 (1983), pp. 777-788), counting anti-chains in a partial order is #P-complete. I don't have access to this paper so I can't tell if this result covers maximal anti-chains or not.