Minimal Surfaces are surfaces which locally minimize area. These surfaces are well-known as mathematical idealizations of soap films, one area of the calculus of variations which applies to geometric modeling. This thesis is devoted to the clas sification of minimal surfaces, specifically limits of minimal surfaces with increasing genus. In this paper, we will see that a particular well-known family of minimal surfaces, indexed by increasing genus, has a limit, and, further, that limit is nearly a well-known example. This is the first nontrivial example of a limit being taken of a family of minimal surfaces of increasing topological complexity. As a classification result, this would limit the set of possible minimal surfaces, as we would see that new surfaces would not be created through the taking of limits of existing families of surfaces in this way.