Examples of non-constructive results I'm giving a talk on constructive mathematics, and I'd like some snappy examples of weird things that happen in non-constructive math.  
For example, it would be great if there were some theorem claiming $\neg \forall x. \neg P(x)$, where no $x$ satisfying $P(x)$ were known or knowable.
But other examples are welcome as well.

The existence of a Hamel Basis, that is, a basis for $\mathbb R$ as a vector space over $\mathbb Q$. No one knows a Hamel basis; it's probably unknowable in some sense.
The existence of a basis for every vector space is equivalent to the axiom of choice, which is the non-constructive piece of math by excellence.
