A long standing question in the study of exotic behavior in dimension four is whether exotic behavior is “stable". For example, in thinking about the four-dimensional h-cobordism theorem, Wall proved that simply connected, exotic four-manifolds always become smoothly equivalent after applying a suitable stabilization operation enough times. Similarly, Hosokawa-Kawauchi and Baykur-Sunukjian showed that exotic surfaces become smoothly equivalent after stabilizing the surfaces some number of times. The question remains, how many stabilizations are necessary, and is one always enough? By considering certain satellite operations, we provide a negative answer to this question in the case of exotic surfaces with boundary. (This draws on joint work with Hayden, Kang, and Park).