The question has a bit of ambiguity in it.

What does it mean that a counterexample cannot be found? What does it mean to find something in mathematics?

We might want to talk about **purely physical computation** (by which I mean using our current technology and reasonably conservative projections thereof; not some hypothetical future in which we have new theorems, algorithms and hardware capable of factoring prime numbers larger than the volume of the observable universe instantly). In this case the word "find" means also verify without any doubt, as well.

In this case we can easily make claims like "Every prime number larger than $2^{2^{100000000000}}$ is of the form $2^n-1$". Of course we can prove that there exists a prime number larger than $2^{2^{100000000000}}$ which is not a Mersenne prime, but computationally speaking verifying that a number which is not a Mersenne prime is prime, is an immensely difficult task.

If the above feels a bit wide, you can also plug in all the other "relatively easily verifiable prime" into the list above. The point in the above example is that there is no efficient way of verifying whether or not a certain number is prime or not; and so to verify whether or not an arbitrary number larger than $2^{2^{100000000000}}$ is a prime number, we are expected to run a *very* lengthy computation. We can replace "prime number" by "solution to sufficiently complicated problem" just as well.

In contrast, we can easily verify if a *given* number is even or not. We just need to check one bit of its binary representation (or one digit of a decimal, or hexadecimal representation), or any other "sufficiently simple problem".

If we talk about **theoretically computation**, it depends on what do you mean prove that a counterexample exists. In particular what do you mean by "prove"? More specifically, prove from what theory?

From $\sf PA$, the axioms of Peano arithmetic, we can develop a nice theory of computation and computability, and we can prove that many natural problems that can be solved, but not in a computable way. For example the ability to decide whether or not a sentence is true or false in the natural numbers.

In this context the term "supercomputer" can be interpreted, perhaps, as an oracle, or more specifically some "additional" function[s] that work in a way we don't necessarily know, and this helps us to solve problems that we couldn't solve before. But even then, we can prove there are always statements that are not computable (now in this stronger sense), but are provably false, so we can't "find" a counterexample.

**We can extend this to mathematical questions** that we can prove existence of certain objects, without our ability to give an explicit construction (read: definition) of an example. For this we usually move one up a notch and use set theory as our theory for the term "prove". Many objects, in particular infinite (and often uncountable) objects are researched in modern mathematics, and we can prove many things about these sets using an axiom known as the Axiom of Choice. This axiom is non-constructive in its nature, as it asserts the existence of certain objects, but doesn't provide us with a way to define them explicitly.

In this context we have so many examples. For example, the existence of non-measurable sets; a linear order of all the sets of sets of natural numbers; a free ultrafilter on the natural numbers.

To sign off this answer, let me point again, that this depends on what you mean "cannot be found" and what it means "prove to exist". In different contexts the answers will differ. And things which may be considered "definable" or considered "found" in one context might not be considered as such in another.