Answer: $9$

Look at the last digits of powers of $9$.

  odd                     even

  $9$                          $81$

$729$                    $ 279\times 9 = ....1$

$ ...1\times9=...9$        $...1$

 $...9$                      $...1$

$9^9$ is a power of $9$ so $9^9$ is odd

This means that $9^{9^9}$ is an odd power of $9$, so it ends in $9$.