group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Given any generalized (Eilenberg-Steenrod) cohomology theory $E$, then for each topological space $X$, there is, by definition, the graded abelian group
This is the $E$-cohomology group of $X$. Now if $E$ is a multiplicative cohomology theory, then these groups inherit the structure of rings. As such
is the $E$-cohomology ring of $X$.
Analogously for various suitable generalizations of the nature of $E$ and $X$ (see at generalized cohomology).