Natural graph class with five excluded subgraphs?
The "split-substitute" graphs are the graphs starting with a split graph and substituting any split graph into any node.
The forbidden induced subgraphs can be obtains from the forbidden induced subgraphs of split graphs (c4, C5, co-C4), but replacing the non-prime ones (c4 and co-c4) with their minimal prime extensions... (here primality is with respect with modular decomposition.)

So the 5 forbidden induced subgraphs are the C5, P5, co-P5, H6 and co-H6. This forms a self-complementary class which strictly contains the class of cographs.

See: http://webdocs.cs.ualberta.ca/~hayward/papers/p4comp.online.pdf



The maxibrittle graphs are the perfectly orderable graphs for which a vertex sequence defined only by the degrees will produce a perfect order (more specifically, the degree sequence defines a brittle ordering). These graphs have exactly 5 minimal forbidden induced subgraphs.