Largest known integer Does there exist a property which is known to be satisfied by only one integer, but such that this property does not provide a means by which to compute this number? I am asking because this number could be unfathomably large. 
I was reading Conjectures that have been disproved with extremely large counterexamples? , does there exist a conjecture that is known to have a counterexample, but which has not been found, and where there is no "bound" on the expected magnitude of this integer?
Is there known something about how the largest integer that is expressable in n symbols, grows with n?

One can easily generate "conjectures" with large counterexamples using Goodstein's theorem and related results. For example, if we conjecture that the Goodstein sequence $\rm\:4_k\:$ never converges to $0$ then the least counterexample is $\rm\ k = 3\ (2^{402653211}-1) \approx 10^{121210695}\:$. For much further discussion of Goodstein's theorem see my sci.math post of Dec 11 1995
