Answer     7

We know that the last digit of $17^k$ repeats as $7$, $9$, $3$, $1$, $7$, $\cdots$. Therefore it is sufficient to compute $17^{17^{17}}\pmod 4$ which is $$17^{17^{17}}\equiv 1^{17^{17}}\equiv 1\pmod 4$$

Therefore, the final answer is $\boxed{7}$.