History-dependent processes are ubiquitous in natural and social systems. Many such processes, especially those that are associated with complex systems, become more constrained as they unfold: The set of possible outcomes of such processes reduces as they ’age’. We demonstrate that stochastic processes of this kind necessarily lead to Zipf’s law in the overall rank distributions of their outcomes. Furthermore, if iid noise is added to such sample-space-reducing processes, the corresponding rank distributions are exact power laws, p(x) ∼ x−λ, where the exponent 0 ≤ λ ≤ 1 directly corresponds to the mixing ratio of process and noise. We illustrate sample-space-reducing processes with an in- tuitive example using a set of dice with different numbers of faces. Sample-space-reducing processes provide a new alternative to understand the origin of scaling in complex systems without the re- course to multiplicative, preferential, or self-organised-critical processes. Potential applications are numerous, ranging from language formation and network growth to statistics of human behaviour.