We study how statistical complexity depends on the system size and how the complexity of the whole system relates to the complexity of its subsystems. We study this size dependence for two well-known complexity measures, the excess entropy of Grassberger and the neural complexity introduced by Tononi, Sporns and Edelman. We compare these results to properties of complexity measures that one might wish to impose when seeking an axiomatic characterization. It turns out that those two measures do not satisfy all those requirements, but a renormalized version of the TSE-complexity behaves reasonably well.