Strong coupling between resonantly matched localized surface plasmons and molecular excitons results in the formation of new hybridized energy states called plexcitons. Understanding the nature and tunability of these hybrid nanostructures is important for both fundamental studies and the development of new applications. We investigate the interactions between J-aggregate excitons and single plasmonic dimers and report for the first time a unique strong coupling regime in individual plexcitonic nanostructures. Dark-field scattering measurements and finite-difference time-domain simulations of the hybrid nanostructures show strong plexcitonic coupling mediated by the near-field inside each dimer gap, which can be actively controlled by rotating the polarization of the optical excitation. The plexciton dispersion curves, obtained from coupled harmonic oscillator models, show anticrossing behavior at the exciton transition energy and giant Rabi splitting ranging between 230 and 400 meV. These energies are, to the best of our knowledge, the largest obtained on individual hybrid nanostructures.