We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewidth, the scramble number is not minor monotone, but it is subgraph monotone and invariant under refinement. We compute the scramble number and gonality of several families of graphs for which these invariants are strictly greater than the treewidth.