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)

The game of Chomp is an example. A short and extremely simple strategy-stealing argument shows that no matter the size of the playing board, the first player always has a winning strategy; the complete argument is given in the Wikipedia article. But there is no general strategy known for an arbitrarily sized board. 
