Frans de Waal transformed the study of animal cognition by showing that many capacities once regarded as uniquely human have deep evolutionary roots shared with other animals. Yet the success of this continuity perspective raises what this article calls the De Waal paradox: if human and nonhuman capacities largely differ in degree and organisation, how can we account for the extraordinary historical trajectory of humans? This article argues that part of the answer may lie in recursive organisation: processes in which outputs become inputs for further applications of the same type of operation. Comparative evidence suggests that recursive capacities occur across nonhuman species, whereas the strongest evidence for open-ended recursion concerns human symbolic and cumulative cultural systems. In particular, human cultural products can become externally stabilised and productively reused across individuals, generations and domains. Technologies become means for producing further technologies, symbolic products enable further symbolic operations and accumulated knowledge becomes a resource for generating new knowledge. Such recursion can amplify relatively modest cognitive differences by transforming not only what is accumulated but also the means of producing what comes next. The framework complements established accounts of cumulative culture, cooperation, teaching and niche construction while generating distinct empirical predictions.