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 $DeltaGamma_{rm X} simeq 0.4$ 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.
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