As mesel said, this is finding same as maximizing $lcm(a_1,..,a_k)$ for $a_i$ such that $a_1+a_2+...+a_k=7$. This follows from the fact that disjoint cycles commute and the order is then just the number of times a cycle must be composed with itself to make each of these disjoint cycles $e$.
But this is a hard question for general $n$- you know that once you have professional mathematicians working to find asymptotics it's a difficult problem.
But for $S_7$, we can see that the highest order is $12$ just by checking the conjugacy classes of n-cycles and fixed-points and finding the largest relatively prime integers. So the answer is $3 \times 4=12$ and we have $(1234)(567)$ as our $\sigma$.
To build some intuition/number sense behind this, we also see that $3$ and $4$ are the integers that are closest to equal and still fit the conditions of adding up to $7$ and being coprime, so their product must be greatest of combinations $a_i$.