Maximal order of an element in a symmetric group

This is Landau's Function.

Asymptotic estimates are known.

André has already provided the name and the link; here's a derivation of the bound $g(n)\lt\mathrm e^{n/\mathrm e}$ in the article. If we could choose all the $l_i:=|\sigma_i|$ freely, only constrained by their sum $n$, we'd want to find the stationary points of the objective function

$$ \prod_il_i-\lambda\sum_il_i\;. $$

Differentiating with respect to $\sigma_j$ yields

$$ \prod_il_i=\lambda l_j\;, $$

so not surprisingly the only stationary point is where all the $l_i$ are equal. Then we can optimize their number $k$ by writing $l_i=n/k$, and we want to maximize

$$ \left(\frac nk\right)^k\;, $$

or equivalently

$$ \log\left(\frac nk\right)^k=k\left(\log n-\log k\right)\;. $$

Taking the derivative with respect to $k$ yields $\log n-\log k=1$ and thus $k=n/\mathrm e$, so ideally we'd want all the $l_i$ to be $\mathrm e$. In that case the product would be $\mathrm e^{n/\mathrm e}$, and the constraints that the $l_i$ have to be coprime integers can only lower that value (quite considerably, as the asymptotic result in the article shows).

This calculation also shows that $\mathrm e$ would be the optimal radix for a Fast Fourier Transform.