A very severe storm in the Antarctic belt is analysed that sent a very large swell throughout the South-Pacific Ocean. The reasons for the storm were a deep depression passing over an anomalous warm sea area, with consequent increased intensity, more active wind input, gustiness, with also dynamical generation. Wind and wave model results are verified with scatterometer and altimeter data. We follow the swell evolution during the five days required to reach the Galapagos Islands and a buoy off the Peruvian coast. The first forerunners peaked at 0.032 Hz at these locations, well represented in the model thanks to a purposely extended frequency range used in the WAM model. A nonlinear combined analysis is carried out to estimate the overall maximum single wave heights that may have impinged on the Galapagos coasts. Single wave heights up to 6 m have been estimated. Once generated, the swell conditions at Galapagos and the buoy are perfectly anticipated. Including generation, useful forecasts extend till at least eight days before the event. The lack of any local communication is discussed. An analysis using ERA5 winds, but a respectively higher resolution long-term wave hindcast, shows that a similar, actually stronger, event happened in 2006. A simple, but sound method, based on physical principles and elementary geometry, is proposed to estimate, firsthand and after any time, the maximum height of a once generated swell. The results for the 2015 storm are correct within 5% of the model values.