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Almost 10 yr of γ-ray observations with the Fermi Large Area Telescope have revealed extreme γ-ray outbursts from flat spectrum radio quasars (FSRQs), temporarily making these objects the brightest γ-ray emitters in the sky. Yet, the location and mechanisms of the γ-ray emission remain elusive. We characterize long-term γ-ray variability and the brightest γ-ray flares of six FSRQs. Consecutively zooming in on the brightest flares, which we identify in an objective way through Bayesian blocks and a hill-climbing algorithm, we find variability on subhour timescales and as short as minutes for two sources in our sample (3C 279 and CTA 102) and weak evidence for variability at timescales less than the Fermi satellite's orbit of 95 minutes for PKS 1510–089 and 3C 454.3. This suggests extremely compact emission regions in the jet. We do not find any signs of γ-ray absorption in the broad-line region (BLR), which indicates that γ-rays are produced at distances greater than hundreds of gravitational radii from the central black hole. This is further supported by a cross-correlation analysis between γ-ray and radio/millimeter light curves, which is consistent with γ-ray production at the same location as the millimeter core for 3C 273, CTA 102, and 3C 454.3. The inferred locations of the γ-ray production zones are still consistent with the observed decay times of the brightest flares if the decay is caused by external Compton scattering with BLR photons. However, the minute-scale variability is challenging to explain in such scenarios. [ABSTRACT FROM AUTHOR]