The iterated Kleene closure of an alphabet

Question

The Kleene closure of an alphabet $A$ is the set of all finite sequences with terms in $A$. However, I am interested in defining something like an "iterated" Kleene closure. Let me give an example to illustrate. Consider the alphabet $\{a,b\}$. Some elements of the iterated Kleene closure are $a$, $b$, $(a,b,b)$, $(a,b,(a,b))$, $(a,(a,(a,a),a),a)$. Basically, you take the Kleene closure of the set and iterate it $\omega$ times. What I am looking for is the formal definition of my concept. Also, has any mathematician defined this concept previously?