Tailoring nanoscale light concentration and electromagnetic near-field enhancement over a broad spectral range is crucial for many photonics applications such as infrared spectroscopy, photodetection, and light harvesting. So far, broadband light enhancement has faced significant challenges due to the difficulty of efficiently exciting resonances at spectrally separated wavelengths and the inability of current devices to individually tune each specific resonance. Here, we introduce a multiresonant structure based on the non-overlapping combination of plasmonic nano antenna arrays with multiple periodicities. The self-similarity of the multiperiodic array, obtained by a fractal-like generation procedure, enables the excitation of a high number of resonances without compromising their excitation efficiency. We experimentally demonstrate devices with up to four independent resonances covering an unprecedentedly wide spectral range from 10 to 1.5 mu m. Significantly, the reflectance signal is uniformly strong for all the resonances, reaching more than 70% amplitude and near-field intensity enhancements above 1000. We further show that each individual resonance wavelength can be independently controlled over a 50% spectral range by modifying a single geometrical antenna parameter, providing superior flexibility in tailoring the overall spectral response. Due to the self-similar layout and independent resonances, our design is well described by temporal coupled-mode theory, allowing for a straightforward extension for other nanophotonic applications. Finally, we demonstrate that the wide spectral coverage of our design enables a unique sensing method by simultaneously performing chemically specific mid-infrared detection and near infrared refractometry.