The results would be a model of type theory (with ?? and ?? but no Id) which models Type : Type. It is not trivial, but all types are inhabited (and have the fixed-point property). We can still interpret Girard's paradox in it, but it is not really a paradox without identity types as the type structure does not collapse. Has something like that known? (I fully expect Thomas to pull something out of his beard.)
Indeed this was done in the Thesis of Nancy J. McCracken in 1979 supervised by John Reynolds. Its title is An investigation of programming languages with a polymorphic type structure. Cardelli and Wegner got interested in this in the mid 1980s. There are 2 paper on this one can find on the web Cardelli & Wegner On Understanding Types Data Abstraction and Polymorphism Computing Surveys, Vol.17 (1985) Cardelli A Polymorphic Lambda Calculus with Type:Type a DEC report from 1986 That's what I have wrung out of my beard Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]