Let $X$ be a sufficiently nice topological space (i.e. an object of a category of spaces where the reduced suspension-loops, $(\Sigma, \Omega)$, holds.)
There are two directed systems of spaces one can consider:
1) $X\to \Omega\Sigma X \to \Omega^2\Sigma^2X \to \cdots$
Here the first arrow is the unit $\eta_X$ of the adjunction, the second is is $\Omega \eta_{\Sigma X}$, etc. We get that the long composition $X\to \Omega^n\Sigma^nX$ is the unit of the adjunction $(\Sigma^n,\Omega^n)$ obtained by composing $n$ times $(\Sigma, \Omega)$ with itself.
2) $X\to \Omega \Sigma X \to \Omega \Sigma \Omega \Sigma X \to \cdots$
Here each arrow is a unit of the adjunction $(\Omega, \Sigma)$, applied to the spaces $X, \Omega\Sigma X, \dots$.
Now, the category of spaces is cocomplete, so we may take colimits. The first one yields the well-known space $QX$, whose homotopy groups are the stable homotopy groups of $X$.
Question: Is there an interesting description of the colimit of the second system? Or of its homotopy groups?
I have asked a question regarding a generalization of these systems here.