A key step in understanding the evolution of human language involves unravelling the origins of language's syntactic structure. One approach seeks to reduce the core of syntax in humans to a single principle of recursive combination, merge, for which there is no evidence in other species. We argue for an alternative approach. We review evidence that beneath the staggering complexity of human syntax, there is an extensive layer of nonproductive, nonhierarchical syntax that can be fruitfully compared to animal call combinations. This is the essential groundwork that must be explored and integrated before we can elucidate, with sufficient precision, what exactly made it possible for human language to explode its syntactic capacity, transitioning from simple nonproductive combinations to the unrivalled complexity that we now have.