  The scramble number of a graph provides a lower bound for gonality and an
upper bound for treewidth, making it a graph invariant of interest. In this
paper we study graphs of scramble number at most two, and give a classification
of all such graphs with a finite list of forbidden topological minors. We then
prove that there exists no finite list of forbidden topological minors to
characterize graphs of any fixed scramble number greater than two.
