We study the behaviour of a natural measure defined on the leaves of the genealogical tree of some branching processes, namely self-similar growth-fragmentation processes. Each particle, or cell, is attributed a positive mass that evolves in continuous time according to a positive self-similar Markov process and gives birth to children at negative jumps events. We are interested in the asymptotics of the mass of the ball centered at the root, as its radius decreases to 0. We obtain the almost sure behaviour of this mass when the Eve cell starts with a strictly positive size. This differs from the situation where the Eve cell grows indefinitely from size 0. In this case, we show that, when properly rescaled, the mass of the ball converges in distribution towards a non-degenerate random variable. We then derive bounds describing the almost sure behaviour of the rescaled mass. Those results are applied to certain random surfaces, exploiting the connection between growth-fragmentations and random planar maps obtained in (Probab. Theory Related Fields 172 (2018) 663???724). This allows us to extend a result of Le Gall (Ann. Inst. Henri Poincar?? Probab. Stat. 55 (2019) 237???313) on the volume of a free Brownian disk close to its boundary, to a larger family of stable disks. The upper bound of the mass of a typical ball in the Brownian map is refined, and we obtain a lower bound as well.