We illustrate the potential of a very high energy lepton collider (from 10 to 30 TeV center of mass energy) to explore new physics indirectly in the vector boson fusion double Higgs production process and in direct diboson production at high energy. Double Higgs production is found to be sensitive to the anomalous Higgs trilinear coupling at the percent level, and to the Higgs compositeness xi parameter at the per mille or sub-per mille level thanks to the measurement of the cross-section in the di-Higgs high invariant mass tail. High energy diboson (and tri-boson) production is sensitive to Higgs-lepton contact interaction operators at a scale of several tens or hundred TeV, corresponding to a reach on the Higgs compositeness scale well above the one of any other future collider project currently under discussion. This result follows from the unique capability of the very high energy lepton collider to measure Electroweak cross-sections at 10 TeV energy or more, where the effect of new physics at even higher energy is amplified. The general lesson is that the standard path towards precision physics, based on measurements of high-statistics processes such as single and double Higgs production, is accompanied at the very high energy lepton collider by a second strategy based on measurements at the highest available energy.