  Large field excursions are required in a number of axion models of inflation.
These models also possess global cosmic strings, around which the axion follows
a path mirroring the inflationary trajectory. Cosmic strings are thus an
interesting theoretical laboratory for the study of transplanckian field
excursions. We describe connections between various effective field theory
models of axion monodromy and study the classical spacetimes around their
supercritical cosmic strings. For small decay constants $f<M_p$ and large
winding numbers $n>M_p/f$, the EFT is under control and the string cores
undergo topological inflation, which may be either of exponential or power-law
type. We show that the exterior spacetime is nonsingular and equivalent to a
decompactifying cigar geometry, with the radion rolling in a potential
generated by axion flux. Signals are able to circumnavigate infinite straight
strings in finite but exponentially long time, $t\sim e^{\Delta a/M_p}$. For
finite loops of supercritical string in asymptotically flat space, we argue
that if topological inflation occurs, then topological censorship implies
transplanckian censorship, or that external observers are forbidden from
threading the loop and observing the full excursion of the axion.
