AbstractIn this study, all available data on the largest solar proton events (SPEs), or extreme solar energetic particle (SEP) events, for the period from 1561 up to now are analyzed. Under consideration are the observational, methodological and physical problems of energy-spectrum presentation for SEP fluxes (fluences) near the Earth's orbit. Special attention is paid to the study of the distribution function for extreme fluences of SEPs by their sizes. The authors present advances in at least three aspects: 1) a form of the distribution function that was previously obtained from the data for three cycles of solar activity has been completely confirmed by the data for 41 solar cycles; 2) early estimates of extremely large fluences in the past have been critically revised, and their values were found to be overestimated; and 3) extremely large SEP fluxes are shown to obey a probabilistic distribution, so the concept of an “upper limit flux” does not carry any strict physical sense although it serves as an important empirical restriction. SEP fluxes may only be characterized by the relative probabilities of their appearance, and there is a sharp break in the spectrum in the range of large fluences (or low probabilities). It is emphasized that modern observational data and methods of investigation do not allow, for the present, the precise resolution of the problem of the spectrum break or the estimation of the maximum potentialities of solar accelerator(s). This limitation considerably restricts the extrapolation of the obtained results to the past and future for application to the epochs with different levels of solar activity.