Prove that a counterexample exists without knowing one

Solution 1:

Statement: "There are no primes greater than $2^{60,000,000}$". No known counter-example. Counter example must exist since the set of primes is infinite (Euclid).

Solution 2:

According the Wikipedia article on Skewes' number, there is no explicit value $x$ known (yet) for which $\pi(x)\gt\text{li}(x)$. (There are, however, candidate values, and there are ranges within which counterexamples are known to lie, so this may not be what the OP is after.)

Another example along the same lines is the Mertens conjecture.

A somewhat silly example would be the statement "$(100!)!+n+2$ is composite." It's clear that $S(n)$ is true for all "small" values of $n\in\mathbb{N}$, and it's clear that it's false in general, but I'd be willing to bet a small sum of money that no counterexample will be found in the next $100!$ years....

(Note: I edited in a "$+2$" to make sure that my silly $S(n)$ is clearly true for $n=0$ and $1$ as well as other "small" values of $n$.)