  We analyse high-quality NuSTAR observations of the local (z = 0.011) Seyfert
2 active galactic nucleus (AGN) IC 3639, in conjunction with archival Suzaku
and Chandra data. This provides the first broadband X-ray spectral analysis of
the source, spanning nearly two decades in energy (0.5-30 keV). Previous X-ray
observations of the source below 10 keV indicated strong reflection/obscuration
on the basis of a pronounced iron fluorescence line at 6.4 keV. The hard X-ray
energy coverage of NuSTAR, together with self-consistent toroidal reprocessing
models, enables direct broadband constraints on the obscuring column density of
the source. We find the source to be heavily Compton-thick (CTK) with an
obscuring column in excess of $3.6\times10^{24}$ cm$^{-2}$, unconstrained at
the upper end. We further find an intrinsic 2-10 keV luminosity of
$\textrm{log}_{10}(L_{\textrm{2-10 keV}} \textrm{[erg s}^{-1}]) =
43.4^{+0.6}_{-1.1}$ to 90% confidence, almost 400 times the observed flux, and
consistent with various multi-wavelength diagnostics. Such a high intrinsic to
observed flux ratio in addition to an Fe-K$\alpha$ fluorescence line equivalent
width exceeding 2 keV is extreme amongst known bona fide CTK AGN, which we
suggest are both due to the high level of obscuration present around IC 3639.
Our study demonstrates that broadband spectroscopic modelling with NuSTAR
enables large corrections for obscuration to be carried out robustly, and
emphasises the need for improved modelling of AGN tori showing intense iron
fluorescence.
