We all know that $a^nu(n)$ has unilateral $Z$-transform. But what is the $Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'.

But bilateral Laplace transform of $e^t$ doesn't exist. Both are exponentials  in discrete and continuous  domain respectively. Considering the similarity between Laplace and $Z$-transform, how to explain the above problem?