Finding the supremum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

You know that as a real function, $x^{1/x}$ is increasing on $(0, e)$ and decreasing on $(e, \infty)$. The same must be true if we consider it a function on the integers. That means that the maximum among integers must be at either $2$ or at $3$ (since any other integer input must give a function value strictly smaller than one of these two). Now just check those two input values, and you're done.