Reaching the quantum optics limit of strong light-matter interactions between a single exciton and a plasmon mode is highly desirable, because it opens up possibilities to explore room-temperature quantum devices operating at the single-photon level. However, two challenges severely hinder the realization of this limit: the integration of single-exciton emitters with plasmonic nanostructures and making the coupling strength at the single-exciton level overcome the large damping of the plasmon mode. Here, we demonstrate that these two hindrances can be overcome by attaching individual J aggregates to single cuboid Au@Ag nanorods. In such hybrid nanosystems, both the ultrasmall mode volume of ∼71 nm^{3} and the ultrashort interaction distance of less than 0.9 nm make the coupling coefficient between a single J-aggregate exciton and the cuboid nanorod as high as ∼41.6 meV, enabling strong light-matter interactions to be achieved at the quantum optics limit in single open plasmonic nanocavities.