  Many cosmological models rely on large couplings of axions to gauge fields.
Examples include theories of magnetogenesis, inflation on a steep potential,
chiral gravitational waves, and chromonatural inflation. Such theories require
a mismatch between the axion field range and the mass scale appearing in the $a
F \widetilde{F}$ coupling. This mismatch suggests an underlying monodromy, with
the axion winding around its fundamental period a large number of times. We
investigate the extent to which this integer can be explained as a product of
smaller integers in a UV completion: in the parlance of our times, can the
theory be "clockworked"? We argue that a clockwork construction producing a
potential $\mu^4 \cos(\frac{a}{j F_a})$ for an axion of fundamental period
$F_a$ will obey the constraint $\mu < F_a$. For some applications, including
chromonatural inflation with sub-Planckian field range, this constraint
obstructs a clockwork UV completion. Alternative routes to a large coupling
include fields of large charge (an approach limited by strong coupling) or
kinetic mixing (requiring a lighter axion). Our results suggest that
completions of axion cosmologies that explain the large parameter in the theory
potentially alter the phenomenological predictions of the model.
