Resolving this question would have broad consequences. A positive answer would give logicians and algebraists a concrete finite basis for reasoning about number-theoretic identities involving exponentiation, with applications to automated theorem proving and formal verification of arithmetic. A negative answer, proving that no finite axiomatization exists, would place the equational theory of the naturals in the same class as other notoriously wild undecidable or non-finitely-based theories, sharpening our understanding of the limits of algebraic reasoning about exponentiation and connecting to deep questions in complexity theory about the expressive power of arithmetic terms.