Many biological signaling pathways employ proteins that competitively dimerize in diverse combinations. These dimerization networks can perform biochemical computations, in which the concentrations of monomers (inputs) determine the concentrations of dimers (outputs). Despite their prevalence, little is known about the range of input-output computations that dimerization networks can perform (their ‘expressivity’) and how it depends on network size and connectivity. Using a systematic computational approach, we demonstrate that even small dimerization networks (3-6 monomers) are expressive, performing diverse multi-input computations. Further, dimerization networks are versatile, performing different computations when their protein components are expressed at different levels, such as in different cell types. Remarkably, individual networks with random interaction affinities, when large enough (>=8 proteins), can perform nearly all (~90%) potential one-input network computations merely by tuning their monomer expression levels. Thus, even the simple process of competitive dimerization provides a powerful architecture for multi-input, cell-type-specific signal processing.