The transmission of a wave through a randomly chosen "pile of plates" typically decreases exponentially with the number of plates, a phenomenon closely related to Anderson localization. In apparent contradiction, we construct disordered planar permittivity profiles which are complex valued (i.e., have reactive and dissipative properties) that appear to vary randomly with position, yet are one-way reflectionless for all angles of incidence and exhibit a transmission coefficient of unity. In addition to these complex-valued "random" planar permittivity profiles, we construct a family of real-valued, two-way reflectionless and perfectly transmitting disordered permittivity profiles that function only for a single angle of incidence and a narrow frequency range.