Fractal-shaped nanoantennas have a large potential to enable multiband devices for surface-enhanced spectroscopy due to their scale-invariant geometry that gives rise to strongly enhanced local fields across different spectral ranges with multiscale spatial distributions. In particular, fractal nanoantennas based on plasmonic metals are promising for biodetection applications that extend from the near-infrared across the mid-infrared spectrum. In this context, we introduce novel multiscale resonant structures based on the inverse Cesaro space-filling fractal curve with the remarkable property that the number of resonant bands does not depend on the overall size of the structures. We systematically study their scattering and near-field resonant properties by resorting to full-field finite difference time domain simulations in combination with experimental Fourier transform infrared microspectroscopy. In particular, by investigating a number of gold antennas fabricated by electron-beam lithography on CaF2 substrates, we demonstrate controllable multiband plasmonic resonances from near infrared to the mid-infrared spectral regions. Moreover, our findings demonstrate that large values of near-field enhancement with hierarchical fractal distributions can be achieved in Cesaro-type nanoantennas across multiple bands that are ideally suited for chemical detection on a small footprint area. In order to demonstrate the full potential of Cesaro fractal nanoantennas for infrared sensing spectroscopy, we show triple band reliable detection of thin poly(methyl methacrylate) layers with nanoscale thickness. The engineering of Cesaro-type plasmonic nanoantennas provides a novel strategy for the realization of active devices with a large spectral density and reduced footprints that can be conveniently integrated in future plasmonic-photonic active platforms for energy harvesting and optical biosensing.