Robertson and Seymour tell us that any minor-closed family of graphs has a finite collection of excluded minors.

Standard examples include planar graphs with two excluded minors ($K_5$ and $K_{3,3}$) and knotlessly embeddable graphs with the Petersen family as excluded minors.

However in general, a finite list is not necessarily a short list, and there are natural properties with large numbers of excluded minors. In fact, I recall seeing examples where astronomical numbers of excluded minors have been proved to exist - unfortunately for the life of me, I can't remember these examples and I can't find them on MathSciNet.

So my question is: please give me references to results that show the existence of vast numbers of excluded minors for natural minor-closed classes of graphs.