Motivated by nucleation and molecular aggregation in physical, chemical and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistic of times it requires for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volume, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volume, we develop a scaling framework to study the behavior of the first assembly time as a function of the total quantity of particles. We find that the mean time to first completion of a maximum-sized cluster may have surprisingly a very weak dependency on the total number of particles. We highlight how the higher statistic (variance, distribution) of the first passage time may still help to infer key parameters (such as the size of the maximum cluster) from data. And last but not least, we present a framework to quantify the formation of cluster of macroscopic size, whose formation is (asymptotically) very unlikely and occurs as a large deviation phenomenon from the mean-field limit. We argue that this framework is suitable to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.