We have investigated the formation of large crystals in the tropical tropopause layer with simulations ranging from simple growth-sedimentation calculations for individual crystals to one-dimensional, trajectory-driven Lagrangian calculations including depletion of water vapor by deposition growth. We find that the large crystals can only be explained if the crystals were grown in an extremely supersaturated environment. A very selective nucleation process would be required such that only a small number of crystals are produced preventing reduction of the supersaturation caused by crystal growth. This conclusion is supported by the observations of large supersaturations with respect to ice (relative humidities with respect to ice as large as about 200%). We will show that these results have significant implications for dehydration of air entering the stratosphere.