Aims. We measure the Crab Nebula γ-ray spectral energy distribution in the ~100 TeV energy domain and test the validity of existing leptonic emission models at these high energies. Methods. We used the novel very large zenith angle observations with the MAGIC telescope system to increase the collection area above 10 TeV. We also developed an auxiliary procedure of monitoring atmospheric transmission in order to assure proper calibration of the accumulated data. This employs recording optical images of the stellar field next to the source position, which provides a better than 10% accuracy for the transmission measurements. Results. We demonstrate that MAGIC very large zenith angle observations yield a collection area larger than a square kilometer. In only ~ 56 h of observations, we detect the γ-ray emission from the Crab Nebula up to 100 TeV, thus providing the highest energy measurement of this source to date with Imaging Atmospheric Cherenkov Telescopes. Comparing accumulated and archival MAGIC and Fermi/LAT data with some of the existing emission models, we find that none of them provides an accurate description of the 1 GeV to 100 TeV γ-ray signal.