  We identify sources with extremely hard X-ray spectra (i.e., with photon
indices of Gamma<0.6 in the 13 sq. deg. NuSTAR serendipitous survey, to search
for the most highly obscured AGNs detected at >10 keV. Eight extreme NuSTAR
sources are identified, and we use the NuSTAR data in combination with lower
energy X-ray observations (from Chandra, Swift XRT, and XMM-Newton) to
characterize the broad-band (0.5-24 keV) X-ray spectra. We find that all of the
extreme sources are highly obscured AGNs, including three robust Compton-thick
(CT; N_H > 1.5e24 cm^-2) AGNs at low redshift (z<0.1), and a likely-CT AGN at
higher redshift (z=0.16). Most of the extreme sources would not have been
identified as highly obscured based on the low energy (<10 keV) X-ray coverage
alone. The multiwavelength properties (e.g., optical spectra and X-ray/MIR
luminosity ratios) provide further support for the eight sources being
significantly obscured. Correcting for absorption, the intrinsic rest-frame
10-40 keV luminosities of the extreme sources cover a broad range, from ~ 5 x
10^42 to 10^45 erg s^-1. The estimated number counts of CT AGNs in the NuSTAR
serendipitous survey are in broad agreement with model expectations based on
previous X-ray surveys, except for the lowest redshifts (z<0.07) where we
measure a high CT fraction of f_CT^obs = 30 (+16 -12) %. For the small sample
of CT AGNs, we find a high fraction of galaxy major mergers (50 +/- 33%)
compared to control samples of "normal" AGNs.
