Are there any non-constructive proofs for which an example was never constructed? By non-constructive I mean the following:
A mathematical object is proven to exist yet it is not constructed in the proof.
Are there any examples of proofs like this where the mathematical object was never constructed? (by which i mean even after the existence of it was proven)

On the same line of thought but, imo, more striking, is the use of Zermelo's Theorem to prove there must exist a well-ordering of the reals (and thus, that golden dream of having a grip on that elusive first positive real number seems to be closer...).
Yet no such ordering on $\;\Bbb R\;$, as far as I am aware, is known. Of course, Zermelo's Theorem, Zorn's Lemma and The Axiom of Choice are all logically equivalent in ZFC.
