Human culture is uniquely cumulative and open-ended. Using a computational model of cultural evolution in which neural network based agents evolve ideas for actions through invention and imitation, we tested the hypothesis that this is due to the capacity for recursive recall. We compared runs in which agents were limited to single-step actions to runs in which they used recursive recall to chain simple actions into complex ones. Chaining resulted in higher cultural diversity, open-ended generation of novelty, and no ceiling on the mean fitness of actions. Both chaining and no-chaining runs exhibited convergence on optimal actions, but without chaining this set was static while with chaining it was ever-changing. Chaining increased the ability to capitalize on the capacity for learning. These findings show that the recursive recall hypothesis provides a computationally plausible explanation of why humans alone have evolved the cultural means to transform this planet.