Nucleation and molecular aggregation are important processes in numerous physical and biological systems. In many applications, these processes often take place in confined spaces, involving a finite number of particles.  We examine the classic problem of homogeneous nucleation and self-assembly by deriving and analyzing a fully discrete stochastic master equation.  By enumerating the highest probability steady-states, we derive exact analytical formulae for quenched and equilibrium mean cluster size distributions. Comparing results with those from mass-action models reveals striking differences between the two corresponding equilibrium mean cluster concentrations. These differences depend primarily on the divisibility of the total available mass by the maximum allowed cluster size, and the remainder. When such mass "incommensurability'' arises, a single remainder particle can "emulsify'' the system by significantly broadening the equilibrium mean cluster size distribution. This discreteness-induced broadening effect is periodic in the total mass of the system but arises even when the system size is asymptotically large, provided the ratio of the total mass to the maximum cluster size is finite.  Our findings define a new scaling regime in which results from classic mass-action theories are qualitatively inaccurate, even in the limit of large total system size.  First passage times to the formation of the largest cluster will also be discussed.