TITLE: Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?
QUESTION [94 upvotes]: In mathematics, is there any conjecture about the existence of an object that was proven to exist but that has not been explicitly constructed to this day? Here object could be any mathematical object, such as a number, function, algorithm, or even proof.

REPLY [5 votes]: stackzebra posted an answer, which has been deleted by a moderator. It's a very simple answer, but, I think, a good one, so I'm going to post it (in a modified form) here: 
It has been proved that there is a prime greater than $10^{10^{100}}$ – indeed, Euclid proved that there are infinitely many such primes – but no example has ever been given. The largest known primes are as infinitesimals compared to this number.