The third root of 2 is a real algebraic number. It is not constructible.
If we had a set generated from all the constructible numbers and all finite roots (not just square roots) would this set be the same as the real algebraic numbers?
Alternatively, is there any real algebraic number that is not constructible and not a finite root of some constructible number?
Absolutely there are! For example, there is no way to express the solutions of $x^5-x+1$ as radical expressions. More simply, I think that $\sqrt[3]{2} + \sqrt{3}$ satisfies your criteria.