Colloidal semiconductor nanoplatelets, in which carriers are strongly confined only along one dimension, present fundamentally different excitonic properties than quantum dots, which support strong confinement in all three dimensions. In particular, multiple excitons strongly confined in just one dimension are free to re-arrange in the lateral plane, reducing the probability for multi-body collisions. Thus, while simultaneous multiple photon emission is typically quenched in quantum dots, in nanoplatelets its probability can be tuned according to size and shape. Here, we focus on analyzing multi-exciton dynamics in individual CdSe/CdS nanoplatelets of various sizes through the measurement of second-, third-, and fourth-order photon correlations. Thus, for the first time, we can directly probe the dynamics of the two, three and four exciton states in the single nanocrystal level. Remarkably, although higher orders of correlation vary substantially among the synthesis' products, they strongly correlate with the value of second order antibunching. The scaling of the higher order moments with the degree of antibunching presents a small yet clear deviation from the accepted model of Auger recombination through binary collisions. Such a deviation suggests that many-body contributions are present already at the level of triexcitons. These findings highlight the benefit of high-order photon correlation spectroscopy as a technique to study multi-exciton dynamics in colloidal semiconductor nanocrystals.