group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A cohomology theory $E$ is called multiplicative if $E^\bullet(X)$ is not just a graded abelian group, but actually a graded ring.
A multiplicative structure on a generalized (Eilenberg-Steenrod) cohomology theory is the structure of a ring spectrum on the spectrum that represents it.
e.g. Lurie 10, lecture 4
In particular every E-∞ ring is a ring spectrum, hence represents a multiplicative cohomology theory, but the converse is in general false.
ordinary cohomology with coefficients in a ring, in particular integral cohomology
etc….
For multiplicative cohomology theories one can consider
See also
Jacob Lurie, A Survey of Elliptic Cohomology - cohomology theories
Jacob Lurie, Chromatic Homotopy Theory 2010, Lecture 4 Complex-oriented cohomology theories (pdf)