jmc

Since $4!=4\times 3\times 2\times 1=24$, we need to evaluate the units digit of 4\ = \underbrace{24^{24^{\cdot^{\cdot^\cdot}}}}_{24}$.Theexponentofthebase$24$isanevennumber,say$2n$.Then,$4\$=24^{2n}=576^n$. The units digit of the product of two numbers with units digit $6$ is equal to $6$. Thus, the desired answer is $\boxed{6}$.