In this work, we describe finite element simulations of the plasmonic resonance (PLR) properties of a self-similar chain of plasmonic nanostructures. Using a broad range of conditions, we find strong numerical evidence that the electric field confinement behaves as (Ξ/λ)(PLR)[proporationality] EFE(-γ), where EFE is the electric field enhancement, Ξis the linear size of the focusing length, and λ is the wavelength of the resonant excitation. We find that the exponent γ is close to 1, i.e. significantly lower than the 1.5 found for two-dimensional nanodisks. This scaling law provides support for the hypothesis of a universal regime in which the sub-optical wavelength electric field confinement is controlled by the Euclidean dimensionality and is independent of nanoparticle size, metal nature, or embedding medium permittivity.