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Nonclassical modifications of plasmon-assisted fluorescence enhancement are theoretically explored by placing dipole emitters at the narrow gaps encountered in canonical plasmonic architectures, namely dimers and trimers of different metallic nanoparticles. Through detailed simulations, in comparison with appropriate analytical modelling, it is shown that, within classical electrodynamics, and for the largely reduced separations explored, fluorescence enhancement factors of the order of 106 can be achieved. This remarkable prediction is mainly governed by the dramatic increase in excitation rate triggered by the corresponding field enhancement inside the gaps. Nevertheless, once nonclassical corrections are included, the amplification factors decrease by up to two orders of magnitude and a saturation regime for narrower gaps is reached, as shown by simulations based on the generalized nonlocal optical response theory. A simple strategy to introduce nonlocal corrections to the analytic solutions, which reproduce the trends of the simulations excellently, is also proposed. It is therefore demonstrated that the nonlocal optical response of the metal imposes more realistic, finite upper bounds to the enhancement feasible with ultrasmall plasmonic cavities, thus providing a theoretical description closer to state of the art experiments. [Phys. Rev. B 96, 085413 (2017) doi:10.1103/PhysRevB.96.085413].