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The source IGR J18245–2452 is the fifteenth discovered accreting millisecond X-ray pulsar and the first neutron star to show direct evidence of a transition between accretion- and rotation-powered emission states. These swings provided the strongest confirmation to date of the pulsar recycling scenario. During the two XMM-Newton observations that were carried out while the source was in outburst in April 2013, IGR J18245–2452 displayed a unique and peculiar X-ray variability. In this work, we report on a detailed analysis of the XMM-Newton data and focus on the timing and spectral variability of the source. In the 0.4–11 keV energy band, IGR J18245–2452 continuously switched between lower and higher intensity states, with typical variations in flux by factor of ~100 on time scales as short as a few seconds. These variations in the source intensity were sometimes accompanied by dramatic spectral hardening, during which the X-ray power-law photon index varied from Γ = 1.7 to Γ = 0.9. The pulse profiles extracted at different count-rates, hardnesses, and energies also showed a complex variability. These phenomena were never observed in accreting millisecond X-ray pulsars, at least not on such a short time-scale. Fast variability was also found in the 5.5 and 9 GHz ATCA radio observations that were carried out for about 6 h during the outburst. We interpret the variability observed from IGR J18245–2452 in terms of a hiccup accretion phase, during which the accretion of material from the inner boundary of the Keplerian disk is reduced by the onset of centrifugal inhibition of accretion, possibly causing the launch of outflows. Changes across accretion and propeller regimes have been long predicted and reproduced by magnetohydrodynamic simulations of accreting millisecond X-ray pulsars, but have never observed to produce as extreme a variability as that shown by IGR J18245–2452.