  We study the three-timescale dynamics of a model that describes the El Ni\~no
Southern Oscillation (ENSO) phenomenon, which was proposed in [A. Roberts, J.
Guckenheimer, E. Widiasih, A. Timmermann, and C. K. Jones, Mixed-mode
oscillations of El Ni\~no--Southern Oscillation, Journal of the Atmospheric
Sciences, 73 (2016), pp. 1755--1766]. While ENSO phenomena are inherently
characterised by the presence of multiple distinct timescales, the above model
has previously been studied in a two-timescale context only. Here, we uncover
the geometric mechanisms that are responsible for complex oscillatory dynamics
in a three-timescale regime, and we demonstrate that the system exhibits a
variety of qualitatively different behaviours, such as mixed-mode oscillation
(MMO) with ``plateaus" -- trajectories where epochs of quiescence alternate
with dramatic excursions -- and relaxation oscillation. The latter, although
emergent also in the two-timescale context in appropriate parameter regimes,
had not been documented previously. Moreover, we show that these mechanisms are
relevant to models from other fields of ecological and population dynamics, as
the underlying geometry is similar to the unfolding of
Rosenzweig--MacArthur-type models in three dimensions.
