If we are only able to use min and max which each take two integer arguments and return the lowest and highest integer, respectively, is it possible to construct a function that will return the i'th largest integer from a set of, say, four integers?
For example, the largest is easy. Given the integers $\{a, b, c, d\}$, finding the largest is expressed as $\max(\max(a,b), \max(c,d))$. A similar example exists for the smallest.
Is it impossible to find the second largest using only max and min?
The second largest number in $\{a,b,c,d\}$ is given by $$\min(\max(a,b,c),\max(a,b,d),\max(a,c,d),\max(b,c,d))$$ since the maximum of the four numbers appears in three of the four $\max$ functions, so the overall minimum comes from the remaining $\max$ function, which takes the maximum of all other values, thus returning the second largest. You can easily reduce the above expression to use only two arguments per $\max,\min$ as you already noticed, but that makes it less clear to read, so I did not do that.
This can easily be generalized, the $k$-th largest number of a finite set $A$ of numbers is given by $$\min\left\{\max(B)\ |\ B\subseteq A, |B|=|A|-k+1\right\}$$