Let $N_n$ be the number of throws before all $n$ dice have shown $6$. Set $m_n := E[N_n]$. Write a recursive formula for $m_n$. Suppose we throw $n$ independent dices. After each throw we put aside the dices
showing $6$ and then perform a new throw with the dices not showing $6$, repeating the process until all dices have shown $6$. Let $N_n$ be the number of throws before all dice have shown $6$. Set $m_n := E[N_n]$ and let $Y$ denote the number of dices not showing $6$ after the first throw.
I've showed that $Y \sim b(n,\frac 5 6)$, because we can consider the throw as $n$ independent trials with succes probability $\frac 5 6$.
Now I must write a recursive formula for $m_n$ and decide $m_1, \ldots, m_5$.
Using the definition of expectation of a discrete RV: 
$E[N_n] = 1 \cdot P(N_n = 1) + 2 \cdot P(N_n = 2) + \ldots$
But this is not very recursive - Any suggestions ?

Two closed formulas, not based on the recursion you suggest:
$$
E(N_n)=\sum_{k\geqslant0}\left(1-\left(1-\left(\frac56\right)^k\right)^n\right)
$$
$$
E(N_n)=\sum_{i=1}^n{n\choose i}\frac{(-1)^{i+1}}{1-\left(\frac56\right)^i}
$$
