Since for any $\epsilon>0~\exists |z|=|\frac{2}{(2k+1)\pi}|<\epsilon$ for some $k$ large enough such that $\tan(1/z)=\pm\infty$ there cannot be a Laurent series at $0$.
Does there need to be a Laurent series converging in a neighborhood of $0$ for it to be an essential singularity? Clearly for any $R>0$ the series does not converge in $\{|z|<R\}$ in this case.
What do we call this type of singularity?
$0$ is not an isolated singularity. There is no disk around $0$ In which the function is analytic except for the origin.