Arithmetic with Limited Exponentiation

We present and analyze a natural hierarchy of weak theories, develop analysis in them, and show that they are interpretable in bounded quantifier arithmetic $\text{I}\Delta_0$ (and hence in Robinson arithmetic Q). The strongest theories include computation corresponding to k-fold exponential (fixed k) time, Weak K\"onig's Lemma, and an arbitrary but fixed number of higher level function types with extensionality, recursive comprehension, and quantifier-free axiom of choice. We also explain why interpretability in $\text{I}\Delta_0$ is so rich, and how to get below it.