There are a number of examples from number theory of finite subsets of $\mathbb{Z}$ whose largest element is unknown. For instance, let $S$ be the set of all positive integers that cannot be written in the form $xy+xz+yz$, where $x,y,z$ are positive integers. It is known that $|S|\leq 19$, and eighteen elements of $S$ are known, the largest being 462. If there is another element $n$, then $n>10^{11}$. Assuming the Generalized Riemann Hypothesis, $n$ does not exist. See http://people.math.sfu.ca/~kkchoi/paper/rep.pdf.