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 $γ$-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 $γ$-ray spectrum during one of the flares was very hard, with an index of $Γ_γ = 1.7 ± 0.1$, which is rarely seen in flat spectrum radio quasars. The lack of concurrent optical variability implies a very high Compton dominance parameter $L_γ/L_{\rm syn} > 300$. 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 $ΔΓ_{\rm X} ≃ 0.4$ at $∼$4 keV. Modeling the broad-band SED during this flare with the standard synchrotron plus inverse Compton model requires: (1) the location of the $γ$-ray emitting region is comparable with the broad line region radius, (2) a very hard electron energy distribution index $p ≃ 1$, (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 $γ$-ray and optical spectra cannot satisfactorily explain the production of X-ray emission. ; Comment: 17 pages, 10 figures, accepted for publication in The Astrophysical Journal