Inspired by P_max, we define and explore the *-versions of various forms of Martin's Maximum. While the *-version of BMM is equivalent to BMM (by a result of Asperó and the speaker), the *-version of full MM might be strictly stronger than MM and in fact the former isn't even known to be consistent. The new result will be the construction of a ZFC-model of the *-version of MM for antichains of size ℵ₂. Doing the same for ℵ₃ is a problem that is wide open and leads to interesting set theoretical questions which are currently explored by several people.