jmc

We can rewrite the expression as $$222,222-(22,222+2,222+222+22+2).$$ We note that the units digit of the addition is $0$ because all the units digits of the five numbers are $2$ and $5\ast 2=10$, which has a units digit of $0$. Now, we have something with a units digit of $0$ subtracted from $222,222$. The units digit of this expression is obviously $2$, and we get $\boxed{2}$ as our answer.