Any reference on solving functional and differenial equations involving power towers of a function?

Question

I am curious if this is covered in any textbooks or papers. I am interested in finding out how to solve functional and differential (even pde) equations involving power towers in them. For example if tow(f(x),n)= g(x) where tow meant a diagonal power tower of f with n determining it's height. Also, can this be solved if some of the f's in the tower (not necessarily all of them) were derivatives , or even partial derivatives in a multivariable case? I'm also interested in what happens if n goes to infinity. Thanks.