A power-free language is characterized by the number of symbols used and a limit on how many times a block of symbols can repeat consecutively. For certain values of these parameters, it is known that the number of legal words grows exponentially fast with respect to length. In the terminology of dynamical systems and ergodic theory, this means that the corresponding shift space has positive topological entropy. We prove that in many cases, this shift space has a unique measure of maximal entropy. The proof uses a weak analogue of Bowen's specification property. The lack of any periodic points in power-free shift spaces stands in striking contrast to other applications of specification-based techniques, where the number of periodic points often has exponential growth rate given by the topological entropy.