<p>We study structural consequences of the existence of a Reinhardt embedding (an embedding of the set-theoretic universe into itself). First, we prove that if delta is a regular cardinal greater than the critical point of the embedding and is fixed by it, then no function from delta to delta in the range of the embedding eventually dominates the embedding restricted to delta. Assuming the existence of such an embedding, we use this to obtain elementary embeddings of initial segments of the universe into themselves exhibiting a local form of extendibility. We then refine results of Gabriel Goldberg on almost supercompact cardinals by proving that certain such cardinals cannot be regular. Finally, we use these ideas to give an alternative proof of Kunen’s Inconsistency Theorem.</p>