Strong-field phenomena in optical nanostructures have enabled the integration of nanophotonics, plasmonics and attosecond spectroscopy. For example, tremendous excitement was sparked by reports of nanostructure-enhanced high-harmonic generation. However, there is growing tension between the great promise held by extreme-ultraviolet and attosecond-pulse generation on the nanoscale, and the lack of successful implementations. Here, we address this problem in a study of highly nonlinear optical processes in gas-exposed bow-tie nanoantennas. We find multiphoton- and strong-field-induced atomic excitation and ionization resulting in extreme-ultraviolet fluorescence, as well as third- and fifth-harmonic generation intrinsic to the nanostructures. Identifying the intensity-dependent spectral fingerprint of atomic fluorescence, we gauge local plasmonic fields. Whereas intensities sufficient for high-harmonic generation are indeed achieved in the near-field, the nanoscopic volume is found to prohibit an efficient conversion. Our results illustrate opportunities and challenges in highly nonlinear plasmonics and its extension to the extreme ultraviolet.