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 of Vershinin .

The calculation you want is described in detail in Joe Neisendorfer's book Algebraic Methods in Unstable Homotopy Theory, Cambridge University Press, 2010. In particular, the Eilenberg--Moore spectral sequence collapses to give that $H_*(\Omega^2 S^3;\mathbb Z) = Cotor_*^{\mathbb Z[y]}(\mathbb Z, \mathbb Z)$. By calculating mod p, and then using the very simple Bockstein spectral sequence, he shows in Corollary 10.26.5, that $p$ annihilates the $p$--torsion for all primes $p$. Thus $H_*(\Omega^2 S^3;\mathbb Z)_{(p)}$ is a graded $\mathbb Z/p$ vector space (above dimension 1) whose Poincare series could easily be worked out from $H_*(\Omega^2 S^3;\mathbb Z/p)$.