smc

Note that every factorial after $5!$ has a unit digit of $0$, meaning that we can disregard them. Thus, we only need to find the units digit of $1!+2!+3!+4!$, and as $1!+2!+3!+4!\equiv 3$ mod $10$, which means that the unit digit is $3$, we have our answer of $\boxed{3}$ as desired.