Which theorems of classical mathematics cannot be proved without using the law of excluded middle? The law of excluded middle is a logical principle that says that for any sentence $A$, the sentence $A\lor\,\neg A$ is true. This is a valid law of classical logic, but is rejected by intuitionistic logic.
However, for some the proofs of mathematical theorems that use the law of excluded middle, there exists an alternative proof of the theorem that does not use the law of excluded middle. 
Is there any theorem of classical mathematics that cannot be proved without using the law of excluded middle?

Well what do you mean by "theorem of classical mathematics"? The answer is going to change depending on how you define it.
If for example you have first-order theory over a language with a predicate symbol $P$ that has no axioms at all. Then $\forall x\ ( P(x) \lor \neg P(x) )$ is a theorem, but is not provable in intuitionistic logic.
For a more crucial example, take any formal system $S$. Then "$S$ is consistent or $S$ is inconsistent." is a classically valid sentence but not provable intuitionistically. So not having classical logic makes one unable to state intuitively true facts.
