I have checked everything "homology of loop spaces"-like, but was not able to find what is $H\_\*(\Omega^2S^3, \mathbb{Z})$. Therefore I ask you how to compute that?



Integer homology of double loop space of odd-dimensional sphere
There are homology isomorphisms $K(Br,1)\to \Omega^2\_0 S^2$ and $\Omega^2 S^3\to \Omega^2\_0 S^2$, so you are really asking about the homology of the stable braid group $Br$ (the colimit of the natural inclusions $Br\_n\hookrightarrow Br\_{n+1}$).


As expected there is no neat description with integral coefficients, but much is known. You'll find a nice summary in Section 4 of [this paper](http://hopf.math.purdue.edu/Vershinin/hobr.pdf) of Vershinin .


