On 2015 June 16, Fermi-LAT observed a giant outburst from the flat spectrum radio quasar 3C 279 with a peak $>100$ MeV flux of $sim3.6times10^{-5};{rm photons};{rm cm}^{-2};{rm s}^{-1}$ averaged over orbital period intervals. It is the historically highest $gamma$-ray flux observed from the source including past EGRET observations, with the $gamma$-ray isotropic luminosity reaching $sim10^{49};{rm erg};{rm s}^{-1}$. During the outburst, the Fermi spacecraft, which has an orbital period of 95.4 min, was operated in a special pointing mode to optimize the exposure for 3C 279. For the first time, significant flux variability at sub-orbital timescales was found in blazar observations by Fermi-LAT. The source flux variability was resolved down to 2-min binned timescales, with flux doubling times less than 5 min. The observed minute-scale variability suggests a very compact emission region at hundreds of Schwarzschild radii from the central engine in conical jet models. A minimum bulk jet Lorentz factor ($Gamma$) of 35 is necessary to avoid both internal $gamma$-ray absorption and super-Eddington jet power. In the standard external-radiation-Comptonization scenario, $Gamma$ should be at least 50 to avoid overproducing the synchrotron-self-Compton component. However, this predicts extremely low magnetization ($sim5times10^{-4}$). Equipartition requires $Gamma$ as high as 120, unless the emitting region is a small fraction of the dissipation region. Alternatively, we consider $gamma$ rays originating as synchrotron radiation of $gamma_{rm e}sim1.6times10^6$ electrons, in magnetic field $Bsim1.3$ kG, accelerated by strong electric fields $Esim B$ in the process of magnetoluminescence. At such short distance scales, one cannot immediately exclude production of $gamma$ rays in hadronic processes.
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