Moduli spaces of geometrically interesting objects are often non-compact. They need to be compactified by adding some degenerate objects. In many cases, this can be done in several ways, leading to a menagerie of birational models, which are related to each other in interesting ways. In this talk, I will explore this idea for the spaces of branched covers of curves, known as the Hurwitz spaces. I will construct a number of compactifications of these spaces by allowing more and more branch points to coincide. I will describe the geometry of the resulting spaces for the case of triple covers.