Are there "essentially non-constructive" statements? There exist constructive and non-constructive proofs.
Sometimes, for a mathematical statement, we can have both non-constructive and a constructive proof. 
However, are there statements for which there is only a non-constructive proof? 
(The fact that there maybe a construction of the required object but we don't know  it yet doesn't count here).
Phrased differently, are there statements (that claim existence of objects) that are essentially non-constructive?
 A: There exist real numbers that don't have names. You can prove this, but you can't construct such a real number, because by the very act of constructing it, you would be giving it a name. 
A: The existence of a basis for $\mathbb  R$ as a vector space over $\mathbb Q$ is a consequence of the axiom of choice but it has no constructive proof.
A: There are various flavors of constructive mathematics. Depending on what you mean exactly there might be different answers. But more importantly you would like to find a statement that makes sense from constructive view-point (and has the same sense from classical viewpoint) that is provable classically but not constructively. For example, if you think about real numbers, you cannot simply talk about arbitrary subsets of real numbers since the are not well-defined constructively. In fact you cannot talk about the set of real numbers as an object like we do in classical mathematics. Moreover different definitions of real numbers are not constructively equivalent.
The example given by Gerry is not true constructively simple because they don't believe in existence of those numbers. Similar problems are involved in lhf's answer.
The simplest example that one can give of a statement that cannot be proven constructively is the principle of excluded middle for arbitrary complex formulas.
A: Two canonical examples: the axiom of choice and the law of excluded middle are both assertions that are provable in a classical setting, but they aren't provable constructively.
