What are some things we can prove they must exist, but have no idea what they are? What are some things we can prove they must exist, but have no idea what they are?
Examples I can think of:


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*Values of the Busy beaver function: It is a well-defined function, but not computable.

*It had been long known that there must be self-replicating patterns in Conway's Game of Life, but the first such pattern has only been discovered in 2010

*By the Well-ordering theorem under the assumptions of the axiom of choice, every set admits a well-ordering. (See also Is there a known well ordering of the reals?)


I'm suprised that the following two haven't shown up:


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*What is the smallest Riesel number?

*What is the smallest Sierpiński number? 


In both cases we know they exist because they are smaller than or equal to 509,203 and 78,557 respectively.
