In this talk, we present quite recent results regarding doubly nonlinear PDEs exhibiting a nonstandard growth, namely, we will focus on the local boundedness for a suitable De Giorgi class of functions and on the expansion of positivity for anisotropic PDEs. These results are the product of a collaboration between S. Ciani (University of Bologna), M. Savchenko and I. Skrypnik (Institute of Applied Mathematics and Mechanics, NASU).