  The well studied blazar Markarian 421 (Mrk 421, $z$=0.031) was the subject of
an intensive multi-wavelength campaign when it flared in 2013 April. The
recorded X-ray and very high energy (VHE, E$>$100 GeV) $\gamma$-ray fluxes are
the highest ever measured from this object. At the peak of the activity, it was
monitored by the hard X-ray focusing telescope {\it Nuclear Spectroscopic
Telescope Array} ({\it NuSTAR}) and {\it Swift} X-Ray Telescope (XRT). In this
work, we present a detailed variability analysis of {\it NuSTAR} and {\it
Swift}-XRT observations of Mrk 421 during this flaring episode. We obtained the
shortest flux doubling time of 14.01$\pm$5.03 minutes, which is the shortest
hard X-ray (3$-$79 keV) variability ever recorded from Mrk 421 and is on the
order of the light crossing time of the black hole's event horizon. A pattern
of extremely fast variability events superposed on slowly varying flares is
found in most of the {\it NuSTAR} observations. We suggest that these peculiar
variability patterns may be explained by magnetic energy dissipation and
reconnection in a fast moving compact emission region within the jet. Based on
the fast variability, we derive a lower limit on the magnetic field strength of
$B \ge 0.73 \delta_1^{-2/3} \, \nu_{19}^{1/3}$~G, where $\delta_1$ is the
Doppler factor in units of 10, and $\nu_{19}$ is the characteristic X-ray
synchrotron frequency in units of $10^{19}$~Hz.
