  We report the results of a multi-band observing campaign on the famous blazar
3C 279 conducted during a phase of increased activity from 2013 December to
2014 April, including first observations of it with NuSTAR. The $\gamma$-ray
emission of the source measured by Fermi-LAT showed multiple distinct flares
reaching the highest flux level measured in this object since the beginning of
the Fermi mission, with $F(E > 100\,{\rm MeV})$ of $10^{-5}$ photons cm$^{-2}$
s$^{-1}$, and with a flux doubling time scale as short as 2 hours. The
$\gamma$-ray spectrum during one of the flares was very hard, with an index of
$\Gamma_\gamma = 1.7 \pm 0.1$, which is rarely seen in flat spectrum radio
quasars. The lack of concurrent optical variability implies a very high Compton
dominance parameter $L_\gamma/L_{\rm syn} > 300$. Two 1-day NuSTAR observations
with accompanying Swift pointings were separated by 2 weeks, probing different
levels of source activity. While the 0.5$-$70 keV X-ray spectrum obtained
during the first pointing, and fitted jointly with Swift-XRT is well-described
by a simple power law, the second joint observation showed an unusual spectral
structure: the spectrum softens by $\Delta\Gamma_{\rm X} \simeq 0.4$ at $\sim$4
keV. Modeling the broad-band SED during this flare with the standard
synchrotron plus inverse Compton model requires: (1) the location of the
$\gamma$-ray emitting region is comparable with the broad line region radius,
(2) a very hard electron energy distribution index $p \simeq 1$, (3) total jet
power significantly exceeding the accretion disk luminosity $L_{\rm j}/L_{\rm
d} \gtrsim 10$, and (4) extremely low jet magnetization with $L_{\rm B}/L_{\rm
j} \lesssim 10^{-4}$. We also find that single-zone models that match the
observed $\gamma$-ray and optical spectra cannot satisfactorily explain the
production of X-ray emission.
