We report the discovery of GRB 221009A, the highest flux gamma-ray burst (GRB) ever observed by the Fermi Gamma-ray Burst Monitor (Fermi-GBM). This GRB has continuous prompt emission lasting more than 600 s, which smoothly transitions to afterglow emission visible in the Fermi-GBM energy range (8 keV-40 MeV), and total energetics higher than any other burst in the Fermi-GBM sample. By using a variety of new and existing analysis techniques we probe the spectral and temporal evolution of GRB 221009A. We find no emission prior to the Fermi-GBM trigger time (t 0; 2022 October 9 at 13:16:59.99 UTC), indicating that this is the time of prompt emission onset. The triggering pulse exhibits distinct spectral and temporal properties suggestive of the thermal, photospheric emission of shock breakout, with significant emission up to ~15 MeV. We characterize the onset of external shock at t 0 + 600 s and find evidence of a plateau region in the early-afterglow phase, which transitions to a slope consistent with Swift-XRT afterglow measurements. We place the total energetics of GRB 221009A in context with the rest of the Fermi-GBM sample and find that this GRB has the highest total isotropic-equivalent energy (E γ,iso = 1.0 × 1055 erg) and second highest isotropic-equivalent luminosity (L γ,iso = 9.9 × 1053 erg s-1) based on its redshift of z = 0.151. These extreme energetics are what allowed us to observe the continuously emitting central engine of Fermi-GBM from the beginning of the prompt emission phase through the onset of early afterglow.