How to solve the recurrence: T(n) = n*T(n-1) + n?
Looks like 
$$
T(n) = n! + \left(\frac{n!}{0!} +\frac{n!}{1!} +\frac{n!}{2!} +\frac{n!}{3!} + \dots + \frac{n!}{(n-2)!} + \frac{n!}{(n-1)!}\right)
$$
satisfies the recurrent relation (here $T(0) = 1$).

From this for example $T(n) \leq n!(1+e)$.