Skewes has proved (without assuming RH) that $\pi(x)<Li(x)$ is violated below $e^{e^{e^{e^{7.705}}}}$ which is clearly a very large number.I was wondering if somewhere else some greater number than Skewes's can be found. 



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> **Question** :Is there any published proof in *number theory* breaking this "large number" record?
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Numbers greater than Skewes's whose existence can be found in number theoretic proofs
The [quantitative version](http://people.maths.ox.ac.uk/greenbj/papers/back-of-an-envelope.pdf) of the [Green--Tao theorem](http://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem) says there is an arithemtic progression of length $k$ in the prime numbers below 
$$2^{2^{2^{2^{2^{2^{2^{100k}}}}}}}$$ 
[These are seven 2s, or in non-rendered form for readability 2^2^2^2^2^2^2^(100k)]


