I will discuss properties of dynamical systems with the almost (weak) specification property. In particular, I will outline a construction of shift spaces with the almost (weak) specification property and multiple measures of maximal entropy (MMEs). Using these examples I will demonstrate that conditions of Climenhaga-Thompson (sufficient conditions for intrinsic ergodicity of every subshift factor of a shift space) are in some sense optimal. The talk is based on the joint work with Piotr Oprocha and Michal Rams.