  The purpose of the present work is twofold. First, we develop the theory of
general self-similar growth-fragmentation processes by focusing on martingales
which appear naturally in this setting and by recasting classical results for
branching random walks in this framework. In particular, we establish
many-to-one formulas for growth-fragmentations and define the notion of
intrinsic area of a growth-fragmentation. Second, we identify a distinguished
family of growth-fragmentations closely related to stable L\'evy processes,
which are then shown to arise as the scaling limit of the perimeter process in
Markovian explorations of certain random planar maps with large degrees (which
are, roughly speaking, the dual maps of the stable maps of Le Gall & Miermont.
As a consequence of this result, we are able to identify the law of the
intrinsic area of these distinguished growth-fragmentations. This generalizes a
geometric connection between large Boltzmann triangulations and a certain
growth-fragmentation process, which was established in arXiv:1507.02265 .
