In natural languages, frequency and compositionality exhibit an inverse relationship: the most frequent words often resist regular patterns, developing idiosyncratic forms. This phenomenon, exemplified by irregular verbs where the most frequent verbs resist regular patterns, raises a compelling question: do artificial communication systems follow similar principles?Through systematic experiments with neural network agents in a referential game setting, and by manipulating input frequency through Zipfian distributions, we investigate if these systems mirror the irregular verbs phenomenon, where messages referring to frequent objects develop less compositional structure than messages referring to rare ones.We establish that compositionality is not an inherent property of the frequency itself and provide compelling evidence that limited data exposure, which frequency distributions naturally create, serves as a fundamental driver for the emergence of compositional structure in communication systems, offering insights into the cognitive and computational pressures that shape linguistic systems.