Answer     4

Where $n \ge 5$, $n!$ must be a multiple of $10$ whose units' digit will be $0$. Therefore, the original question is equivalent to finding the unit digit of $$0! + 1! + 2! + 3! + 4! \equiv 1 + 1 + 2 + 6 + 24 \equiv 34\equiv \boxed{4}\pmod{10}$$