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Anderson localization is an interference effect crucial to the understanding of waves in disordered media. However, localization is expected to become negligible when the features of the disordered structure are much smaller than the wavelength. Here we experimentally demonstrate the localization of light in a disordered dielectric multilayer with an average layer thickness of 15 nanometers, deep into the subwavelength regime. We observe strong disorder-induced reflections that show that the interplay of localization and evanescence can lead to a substantial decrease in transmission, or the opposite feature of enhanced transmission. This deep-subwavelength Anderson localization exhibits extreme sensitivity: Varying the thickness of a single layer by 2 nanometers changes the reflection appreciably. This sensitivity, approaching the atomic scale, holds the promise of extreme subwavelength sensing.