There are lots of examples. They even have a name for when a graph has only surjective endomorphisms, such a graph is called a core. In fact every graph is homomorphically equivalent (has a homomorphism both to and from) a unique core. A good reference for cores is Godsil and Royle's Algebraic Graph Theory. Also, Hahn and Tardif's survey on graph homomorphisms is a good reference and it is available online: http://www.mast.queensu.ca/~ctardif/articles/ghss.pdf
More specifically, you could consider the Kneser graphs, which includes the Petersen graph.
Also, I recently proved that all strongly regular graphs are pseudocores: every endomorphism is either surjective or is a mapping to a max clique (http://arxiv.org/abs/1601.00969). From here it follows that these graphs are cores unless they have clique number equal to chromatic number (in which case both are equal to the Hoffman bound/Lovasz theta of the complement since they are SRGs). In practice it seems that almost all are cores. I tested around 80,000 SRGs from Ted Spence's webpage and all but 79 were cores.