  The graph minor structure theorem by Robertson and Seymour shows that every
graph that excludes a fixed minor can be constructed by a combination of four
ingredients: graphs embedded in a surface of bounded genus, a bounded number of
vortices of bounded width, a bounded number of apex vertices, and the
clique-sum operation. This paper studies the converse question: What is the
maximum order of a complete graph minor in a graph constructed using these four
ingredients? Our main result answers this question up to a constant factor.
