Higman constructed an example of a finitely presented simple group here:
A finitely generated infinite simple group.
J. London Math. Soc. 26 (1951), 61–64.
It is a quotient of what is called Higman’s group, which is an amalgamated free product.
Question: is either group isomorphic to a (non-trivial) free product? (i.e. amalgamated over the trivial group)