jmc

First, observe that the units digit of $22^n$ is the same as the ones digit of $2^n$ for all positive integers $n$. Also, observe that the ones digits of $2^1,2^2,2^3,\dots$ are 2, 4, 8, 6, 2, 4, 8, 6, .... Since $22(11^{11})$ is even but not divisible by 4, it leaves a remainder of 2 when divided by 4. Therefore, the units digit of $22^{22(11{)}^{11}}$ is the second digit in the repeating block 2, 4, 8, 6, which is $\boxed{4}$.