  In this Rapid Communication, we describe how the presence of the third
dimension may break the scale invariance in a two-dimensional Bose gas in a
pancake-shaped trap. From the two-dimensional perspective, the possibility of a
weak spilling of the atomic density beyond the ground-state of the confinement
alters the two-dimensional chemical potential; in turn, this correction no
longer supports scale invariance. We compare experimental data with numerical
and analytic perturbative results and find a good agreement.
