You can think of it as the cohomology of the simplicial manifold $X\leftleftarrows X\times G \cdots$ where the $n$-simplices are $X\times G^n$ and the face maps either act on $X$ or multiply two consecutive entries.
Of course, some people will tell you that that is really the same as the Borel construction, but if you're willing to interpret things that liberally, you'll never get away from the Borel construction.