Grothendieck spectral sequence given functors $F,G$, left exact, with as good properties as you want we have a spectral sequence $R^p F\circ R^q G$ abutting to $R^{p+q}(F\circ G)$. I am looking for an analogous for a "mixed version" in te following case: $F$ left exact and $G$ right exact. What appens to $L^pG\circ R^q F$? 
 A: It's only going to work in trivial cases, when objects are both injective and projective, like the category of finite dimensional vector spaces over a field, where your functors are probably exact anyway.  The reason is that in order for the GSS to work, you need your right exact functor $G$ to take projective objects to $F$-acyclic ones, which are typically injective for a left exact functor $F$.  The reason is that you compute $L^iG(A)$ by taking a projective resolution $P^* \rightarrow A$.  Then you hit it with $F$ and you get $F(P^*) \rightarrow F(A)$ (assuming covariance), and you want to make sure you've not created any new homology, so $F(P^*)$ has to be an acyclic complex you can use to compute $R^iF(A)$. 
The problem is that the composition $F \circ G$ has to be either left exact or right exact if you want to say something about its derived functor homology.  And I'm thinking that if $F \circ G$ is left exact, then $G$ has to be left exact too (and hence exact, so all derived functors are 0).  Similarly if $F \circ G$ is right exact, then $F$ has to be right exact too (and hence exact), so everything becomes trivial.
