A while ago I worked on the question of what we can say if $d^n=0$, but I got distracted by more concrete problems. A few people have certainly thought about this question. One place to start looking is "$d^N=0$: generalized homology" or "Generalized homologies for $d^N=0$ and graded $q$-differential algebras" both by Michel Dubois-Violette.
(Sorry for the lack of links; I'm off-campus so I can't actually get to the MathSciNet entries right now.)
