  We construct a family of shift spaces with almost specification and multiple
measures of maximal entropy. This answers a question from Climenhaga and
Thompson [Israel J. Math. 192 (2012), no. 2, 785--817]. Elaborating on our
examples we also prove that some sufficient conditions for every subshift
factor of a shift space to be intrinsically ergodic given by Climenhaga and
Thompson are in some sense best possible, moreover, the weak specification
property neither implies intrinsic ergodicity, nor follows from almost
specification. We also construct a dynamical system with the weak specification
property, which does not have the almost specification property. We prove that
the minimal points are dense in the support of any invariant measure of a
system with the almost specification property. Furthermore, if a system with
almost specification has an invariant measure with non-trivial support, then it
also has uniform positive entropy over the support of any invariant measure and
can not be minimal.
