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)

There are many. In the proof that every vector space has a basis; in the proof that every ideal can be extended to a maximal ideal; in the proof that every filter can be extended to an ultrafilter.
All those use Zorn's lemma and in fact just show that such object exists. None of these proofs can construct the actual object in explicit details, because it is consistent that Zorn's lemma is false and these sort of objects may not exist sometimes (that is, some vector spaces won't have a basis, and so on). Generally speaking the axiom of choice (which is equivalent to Zorn's lemma) is responsible for a lot of the non-constructiveness of modern mathematics, as the axiom asserts the existence of certain objects which cannot be proved to exist otherwise.
If you want another semi-non constructive proof you can use the Cantor-Bernstein theorem for that. It's very easy to prove there is a bijection between $\Bbb{N^N}$ and $\mathcal P(\Bbb N)$ without constructing one, if you are using Cantor-Bernstein. While it is possible to actually construct such bijection, it's much easier to just prove it exists by other means.
In another direction one can argue, for example, that the proof that a continuous function from a closed and bounded interval to $\Bbb R$ must attain its minimum and maximum is non-constructive, because we don't actually construct the point, we just prove it exists.
