Super-strongly connected components?
This is called the Minimal Clique Cover Problem, and is NP-hard: as a matter of fact, the decision version ("can I do it with only $k$ subgraphs?") is one of Karp's original 21 Problems, the ones that first defined NP-completeness.

Since it's linked to graph coloring, you can't even get a good approximation in polynomial time, unfortunately: everything about this problem is hard.

For further reading, your "super-strongly connected components" are generally called cliques, and a clique cover is a way to "cover" the entire graph with non-overlapping cliques. A minimal clique cover uses the smallest possible number of cliques to do it.