  The nonlinear lattice---a new and nonlinear class of periodic
potentials---was recently introduced to generate various nonlinear localized
modes. Several attempts failed to stabilize two-dimensional (2D) solitons
against their intrinsic critical collapse in Kerr media. Here, we provide a
possibility for supporting 2D matter-wave solitons and vortices in an extended
setting---the cubic and quintic model---by introducing another nonlinear
lattice whose period is controllable and can be different from its cubic
counterpart, to its quintic nonlinearity, therefore making a fully `nonlinear
quasi-crystal'.
  A variational approximation based on Gaussian ansatz is developed for the
fundamental solitons and in particular, their stability exactly follows the
inverted \textit{Vakhitov-Kolokolov} stability criterion, whereas the vortex
solitons are only studied by means of numerical methods. Stability regions for
two types of localized mode---the fundamental and vortex solitons---are
provided. A noteworthy feature of the localized solutions is that the vortex
solitons are stable only when the period of the quintic nonlinear lattice is
the same as the cubic one or when the quintic nonlinearity is constant, while
the stable fundamental solitons can be created under looser conditions. Our
physical setting (cubic-quintic model) is in the framework of the
Gross-Pitaevskii equation (GPE) or nonlinear Schr\"{o}dinger equation, the
predicted localized modes thus may be implemented in Bose-Einstein condensates
and nonlinear optical media with tunable cubic and quintic nonlinearities.
