This question was posed by A Rock and a Hard Place in this discussion, where they mentioned the isomorphisms \begin{align*} \mathrm{L}\,\mathrm{Sym}^n_R(M[1]) &\cong (\mathrm{L}\,{\textstyle\bigwedge}^n_R (M))[n],\\ \mathrm{L}\,{\textstyle\bigwedge}^n_R(M[1]) &\cong (\mathrm{L}\,\Gamma^n_R(M))[n]. \end{align*}
From what I understand, these were first proved independently by Bousfield in Homogeneous functors and their derived functors and by Quillen in On the (co)-homology of rings. Two other references are 7.2.1 of Quillen's Homology of Commutative Rings and I.4.3.2.1 (p. 111) of Illusie's Complexe Cotangent et Deformations I.
Question. What comes after divided power algebras in the sequence $\mathrm{Sym}^\bullet_R(M)$, $\bigwedge^\bullet_R(M)$, $\Gamma^\bullet_R(M)$, $\ldots$, in the sense of there existing also algebras $\mathsf{A}^\bullet_\mathsf{3}$, $\mathsf{A}^\bullet_{\mathsf{4}}$, $\ldots$, together with isomorphisms of the form
$$\mathrm{L}\,\mathsf{A}^n_\mathsf{k}(M)\cong (\mathrm{L}\,\mathsf{A}^{n+1}_{\mathsf{k}}(M))[n],$$
where $\mathsf{A}^\bullet_\mathsf{0}\cong\mathrm{Sym}^\bullet_R$, $\mathsf{A}^\bullet_\mathsf{1}\cong\bigwedge^\bullet_R$, and $\mathsf{A}^\bullet_\mathsf{2}\cong\Gamma^\bullet_R$?
One hint was given by Sanath here: the algebra $\mathsf{A}^\bullet_{\mathsf{n}}$ is related to the homology of $\mathrm{K}(\mathbb{Z},n)$, some computations of which may be found in the table in page 67 of Breen–Mikhailov–Touzé's Derived functors of the divided power functors.
