Elementary Functions as a Solvable Group

I am not so happy with the defintion for a elementary function viewed as a finite number of combinations of addition, subtraction, multiplication, division and roots.

It sounds familar to the meaning of a "solutions by radicals" of a polynomial. Though it does has an alternate defintion, it is defined as a solvable group. Is there a way to define these functions also as some solvable group? The only thing I can think of is that the number of solutions to a polynomial is finite. Where the number of elementary functions is not. Thus, the compositions series does not exist. But if you can find one will it not make the proves based on showing there is no such elementary function simpler?

This is my 31:):)th Post!!!