group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A higher chromatic Chern character for Morava E-theory.
The construction in (Torii 03) produces a map from Morava E-theory at chromatic level $(n+1)$ to Morava K-theory of chromatic level $n$, tensored over something. This is induced by degeneration of the corresponding formal group laws.
Matthew Ando, Jack Morava, Hal Sadofsky, Completions of Z/(p)-Tate cohomology of periodic spectra, Geom. Topol. 2 (1998), 145-174 (arXiv:math/9808141)
Takeshi Torii, On degeneration of one-dimensional formal group laws and applications to stable homotopy theory, Amer. J. Math. 125 (2003) 1037–1077 (web)