The article presents the results of applying the method of almost-periodic analysis based on the shift function to the processing of a time series characterizing the index of accumulated cyclone energy in the Atlantic Ocean. Accumulated cyclone energy is an index used to measure the activity of the hurricane season. It combines the number of hurricanes that occurred during the study period, how long they existed, and how powerful were. The studied data are represented by a time series of observations from 1851 to 2023 with a measurement frequency of one year. The results obtained in the course of the study show that the greatest cyclone activity occurs with a periodicity of 15 years and 62 years. The identified almost-periods were confirmed for typhoons in the western Pacific Ocean. The study showed the possibility of estimating the occurrence of extreme annual values of accumulated typhoon energy.