  The trans-series completion of perturbative series of a wide class of quantum
mechanical systems can be determined by combining the resurgence program and
extra input coming from exact WKB analysis. In this paper, we reexamine the
Harper-Hofstadter model and its spectrum, Hofstadter's butterfly, in light of
recent developments. We demonstrate the connection between the perturbative
energy series of the Harper-Hofstadter model and the vev of $1/2$-BPS Wilson
loop of 5d SYM and clarify the differences between their non-perturbative
corrections. Taking insights from the cosine potential model, we construct the
full energy trans-series for flux $\phi=2\pi/Q$ and provide numerical evidence
with remarkably high precision. Finally, we revisit the problem of
self-similarity of the butterfly and discuss the possibility of a completed
version of the Rammal-Wilkinson formula.
