The cumulative open-endedness of human cultures represents a major break with the social traditions of nonhuman species. As traditions are altered and the modifications retained along the cultural lineage, human populations are capable of producing complex traits that no individual could have figured out on its own. For cultures to produce increasingly complex traditions, improvements and modifications must be kept for the next generations to build upon. High-fidelity transmission would thus act as a ratchet, retaining modifications and allowing the historical buildup of complex traditions. Mechanisms acting against slippage are important, of course, but cultures also need to move forward for the ratchet to retain anything important. In this article, I argue that studies of modification-generating processes and the many ways they shape cumulative culture have been overlooked. Key to a better understanding of cultural modification processes is taking seriously that cultural traditions consist of complex, hierarchically structured recipes. Taking such structures seriously and assessing the different ways they can vary in cultural design space, a novel picture for the onset of cumulative cultural evolution emerges. I argue that a possible impediment for cumulative culture in nonhuman animals may in fact reside not so much in the fidelity of their social transmission but rather in the constraints, internal and external, on their capacity to modify complex, hierarchically structured cultural recipes.