  Stack-triangulations appear as natural objects when one wants to define some
increasing families of triangulations by successive additions of faces. We
investigate the asymptotic behavior of rooted stack-triangulations with $2n$
faces under two different distributions. We show that the uniform distribution
on this set of maps converges, for a topology of local convergence, to a
distribution on the set of infinite maps. In the other hand, we show that
rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on
metric spaces to the continuum random tree introduced by Aldous. Under a
distribution induced by a natural random construction, the distance between
random points rescaled by $(6/11)\log n$ converge to 1 in probability.
  We obtain similar asymptotic results for a family of increasing
quadrangulations.
