In this dissertation, we will focus our attention on the limiting behavior of a sequence of compatible orbits of Koch snowflake prefractal billiards. We give a number of theoretical results. In one instance, we construct a well-defined billiard orbit of the Koch snowflake fractal billiard. In another instance, we construct what we call a nontrivial polygonal path. Such a path is constructed from a sequence of basepoints from orbits of prefractal billiards that converges to what we call an elusive limit point of the Koch snowflake. Many other results are concerned with particular properties of particular types of sequences of compatible orbits. One major result is a topological dichotomy for sequences of compatible orbits. A number of figures are used to illustrate the various definitions and concepts. We discuss experimental results indicating the possible existence of a well-defined law of reflection (or some appropriate analogue) for the Koch snowflake fractal billiard. Future directions for research are discussed and a number of conjectures and open questions are given.