Let $Q$ denote $\Omega^\infty\circ \Sigma^\infty$ the free infinite loop space functor. Given some space $X$, we see that $QX$ carries all the stable homotopy information about $X$. Naturally I wanted to do this again $QQX$, but examples such as spheres and eilenberg-maclane spaces get too complicated.

There is a natural maps $\iota :X \to QX$ which means we can have a tower:

$$X \to QX \to Q^2X \to \cdots$$

Taking the colimit I can imagine there being an object $Q^\infty X$. So my questions are:

What is known about repeated applications of the free infinite loop space functor?

Can it be iterated as I have described? What does this describe?