  A central object in any axionic theory is its periodic potential, which is
typically generated by instantons. The goal of this paper is to understand what
physically happens to the theory when we lose control of the potential's
instanton expansion. We argue, using the Yang-Lee theory of phase transitions,
that the theory breaks down in the classic sense: states become light. However,
these states are not necessarily light for all values of the axion and there
can be large regions where the effective description remains valid. We find
alternative expressions for the effective potential in terms of the properties
of these light states, which remain useful even when the instanton expansion
breaks down, and thus initiate a push beyond the lamppost of large instanton
actions. Most of these questions are motivated by the axionic Weak Gravity
Conjecture, which we reformulate without reference to instanton actions. We
also comment on its ability to constrain large-field axion inflation.
