  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
Bertoin et al. [6].
  This allows us to extend a result of Le Gall [24] 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.
