Satellite and ground-based observations are used to explore the composite oceanic–atmospheric link known as the El Niño/La Niña Southern Oscillation (ENSO) phenomenon, which is closely associated with extreme weather events (e.g. heat waves, tornadoes, floods, and droughts), incidence of epidemic diseases (e.g. malaria), severe coral bleaching, etc. The ENSO temporal evolution depends on the energy exchange within the coupled ocean/atmosphere system. Its cycle has an average period of about 4 years, but there is considerable modulation of it from several sources and this is not yet fully understood. In this article, we attempted to explore the intrinsic features of the Best ENSO Index (BEI) from 1870 to 2017. Studying the distribution that characterizes BEI increments, the asymptotic power-law scaling was revealed in their extreme fluctuations. Additionally, in order to study the evolution of BEI data over time, the detrended fluctuation analysis was used, which showed positive long-range correlations of power-law type and multifractal behaviour. These results aim to give a better insight into the global signature of ENSO evolution, considering both the continuous natural interactions taking place between the oceans and the atmosphere and anthropogenic effects. Furthermore, the results obtained could be employed to elucidate the development of more accurate advanced modelling of ocean–atmosphere interactions, thereby improving climate change projections.