We study periodic homogenization and $3D$-$2D$ dimension reduction by $Gamma (pi )$-con-vergence of heterogeneous thin films whose the stored-energy densities have no polynomial growth.In particular, our Drinks results are consistent with one of the basic facts of nonlinear elasticity, Wooden Dolls namely the necessity of an infinite amount of energy to compress a finite volume of matter into zero volume.However, our results are not consistent with the noninterpenetration of the matter.