I am looking for a reference to answer the question in the title. Let me try to clarify a little what I mean:
If a single sheaf $\mathscr F$ has a resolution $\mathscr G^\bullet$ by not necessarily injective objects, then the usual cohomology of $\mathscr F$ is isomorphic to the hypercohomology of $\mathscr G^\bullet$: $$ H^i(X, \mathscr F) \cong \mathbb H^i(X,\mathscr G^\bullet). $$
Now, if one was starting with a complex of sheaves $\mathscr F^\bullet$ and a "resolution" thereof, i.e. a complex of complexes $(\mathscr G^\bullet)^\bullet$, then one should touch on a concept that could be called hyper-hypercohomology.
Yet, I never heard of its existence and I'm pretty sure it does not give you anything new, as soon as you work in the derived category. I just find myself unable to pin down why exactly this is the case.
Any ideas anyone?