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 $\sim3.6\times10^{-5}\;{\rm photons}\;{\rm cm}^{-2}\;{\rm s}^{-1}$ 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 ($\sim5\times10^{-4}$). 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 $\gamma_{\rm e}\sim1.6\times10^6$ electrons, in magnetic field $B\sim1.3$ kG, accelerated by strong electric fields $E\sim B$ in the process of magnetoluminescence. At such short distance scales, one cannot immediately exclude production of $\gamma$ rays in hadronic processes.
