Behavioral economists have attempted to show that human iterated reasoning faculties discourage non-equilibrium, non-convergent game dynamics. But what if individuals iterating through each other's strategic intentions are instead driving complex collective dynamics? The results in this manuscript demonstrate that bounded "what you think I think you think" reasoning can cause sustained deviations from Nash equilibrium and other fixed-point solution concepts. Supporting my thesis are a series of six experiments, a revisitation of a classic game theory experiment, a variety of computational models, and an analysis of a real-world dataset with highly motivated agents. I also introduce two new games, the Mod Game and the Runway Game. By bridging human higher-level reasoning and animal collective behavior, this work challenges attitudes in economics that complex social dynamics can--or even should--be designed away.