  A result of Baykur-Sunukjian states that homologous surfaces in a 4-manifold
become isotopic after a finite number of internal stabilizations, i.e.
attaching tubes to the surfaces. A natural question is how many stabilizations
are needed before the surfaces become isotopic. In particular, given an exotic
pair of surfaces, is a single stabilization always enough to make the pair
smoothly isotopic? We answer this question by studying how the stabilization
distance between surfaces with boundary changes with respect to satellite
operations. Using a range of Floer theoretic techniques, we show that there are
exotic disks in the four-ball which have arbitrarily large stabilization
distance, giving the first examples of exotic behavior in the four-ball for
which "one is not enough".
