Processes occurring in the strong field regime of QED are characterized by background electromagnetic fields of the order of the critical field F_{cr}=m^{2}c^{3}/ℏ|e| in the rest frame of participating charges. It has been conjectured that if in their rest frame electrons and positrons experience field strengths of the order of F_{cr}/α^{3/2}≈1600F_{cr}, with α≈1/137 being the fine-structure constant, their effective coupling with radiation becomes of the order of unity. Here we show that channeling radiation by ultrarelativistic electrons with energies of the order of a few TeV on thin tungsten crystals allows us to test the predictions of QED close to this fully nonperturbative regime by measuring the angularly resolved single photon intensity spectrum. The proposed setup features the unique characteristics that essentially all electrons (1) undergo at most a single photon emission and (2) experience at the moment of emission and in the angular region of interest the maximum allowed value of the field strength, which at 2 TeV exceeds F_{cr} by more than 2 orders of magnitude in their rest frame.
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