I am assuming that you are talking about two-dimensional supersymmetry. While I haven't seen the usage $(2,2)^*$ -- my guess is that it refers to a theory with higher supersymmetry, say $(4,4)$ that is broken to $(2,2)$ by adding some terms that explicitly break it to $(4.4)$. In other words, the spectrum of fields is that of $(4,4)$ but the interactions are $(2,2)$. These are special cases of $(2,2)$ theories since not all $(2,2)$ theories arise in this way.

**Remark: **Many $(2,2)$ (resp. $(4,4)$)) theories arise via dimensional reduction of four-dimensional $\mathcal{N}=1$ (resp. $\mathcal{N}=2$) supersymmetric theories.