Gap-plasmons (GP) in metal-insulator-metal (MIM) structures have shown exceptional performance in guiding and concentrating light within deep subwavelength layers. Reported designs to date exploit tapered thicknesses of the insulating layer in order to confine and focus the GP mode. Here, we propose a mechanism for the three-dimensional concentration of light in planar MIM structures, which exploits exclusively the lateral tapering of the front metallic layer while keeping a constant thickness of the insulating layer. We demonstrate that an array of tapered planar GP nanocavities can efficiently concentrate light in all three dimensions. A semi-analytical, one-dimensional model provides understanding of the underlying physics and approximately predicts the behavior of the structure. Three-dimensional simulations are then used to precisely calculate the optical behavior. Cavities with effective volumes as small as 10-5k3 are achieved in an ultrathin MIM configuration. Our design is inherently capable of efficiently coupling with free-space radiation. In addition, being composed of two electrically continuous layers separated by an ultrathin dielectric spacer, it could find interesting applications in the area of active metamaterials or plasmonic photocatalysis where both electrical access and light concentration are required.