In Wikipedia the hyper-transcendental numbers are defined as complex numbers that are not the value, at an algebraic point, of a function which is the solution of an algebraic differential equation with integer coefficients and with algebraic initial conditions. But there are not given examples of such numbers, and it not specified if such numbers are computable or not.
I can suppose that such numbers can be the values of a hyper-transcendental function (as the gamma function) at some point, but how can we proof that such a number is not the value also of some elementary transcendental function?
Searching on the web I have not found much about such numbers. Are we sure that they exists? There is some known example?