Hence the coefficientcohomology ring of stable homotopy homology theory (its value on the point) is the stable homotopy groups of spheres. This highlights that stable homotopy homology of any space $X$ is extremely hard, or impossible, to completely analyze, since this is true already for the coefficient ring over the point.

Beware that the term stable homotopy theory, which would seem to be the canonical name for this generalized homology theory, traditionally refers instead to the general homotopy theory of spectra. The full term stable homotopy homology theory is used for emphasis, but clunky in practice.