The disappearance of the accretion disc in low-luminosity active galactic nuclei (LLAGN) leaves behind a faint optical nuclear continuum whose nature has been largely debated, mainly due to serious observational limitations in the IR to UV range. We combine multi-wavelength sub-arcsecond resolution observations -able to isolate the genuine nuclear continuum- with nebular lines in the mid-IR, to indirectly probe the shape of the extreme UV continuum. We found that 8 of the nearest prototype LLAGN are compatible with pure compact jet emission (self-absorbed synchrotron plus the associated self-Compton component) over more than ten orders of magnitude in frequency. When compared with typical radio galaxies, the LLAGN continua show two peculiarities: i) a very steep spectral slope in the IR-to-optical/UV range (-3.7<a0<-1.3; F_nu_~nu^a0^); and ii) a very high turnover frequency (0.2-30THz; 1.3mm-10-micron). These attributes can be explained if the synchrotron continuum is mainly dominated by thermalised particles at the jet base or corona with considerably high temperatures, whereas only a small fraction of the energy (~20%) would be distributed along the high-energy power-law tail of accelerated particles. On the other hand, the nebular gas excitation in LLAGN is in agreement with photo-ionisation from inverse Compton radiation (ax~-0.7), which would dominate the nuclear continuum shortwards of ~3000{AA}. Our results suggest that the LLAGN continuum can be dominated at all wavelengths by undeveloped jets, powered by a thermalised particle distribution, similar to the case of compact jets in quiescent black hole X-ray binaries. This has important implications in the context of galaxy evolution, since LLAGN may represent a major but underestimated source of kinetic feedback in galaxies.