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A recent question asked whether the infinite nested logarithm $\ln(2\ln(3\ln(4\ln(5\ln(6…)))))$ has a finite value. It does: I showed with a crude method that the value was at most $2$, and Hagen von Eitzen found a more sophisticated method that gives arbitrarily precise bounds to get an approximate value of $1.36790126$.

Is there any closed form for this expression, or even a form in terms of simple special functions and infinite sums? I tried an inverse symbolic calculator, but it came up short; I also speculated that it might be some special value of the Lambert W function, but it doesn't seem to be, although $W(2e) \approx 1.3748$ isn't far off.

Connor Harris
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    Honestly, I don’t believe a closed form exist. – Lee May 10 '18 at 15:33
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    There is no reason to expect simple value of Lambert, say $W(a)$, which would imply $a=\ln(2\ln(3\ln(4\ln(5\ln(6…)))))2\ln(3\ln(4\ln(5\ln(6…))))$. –  May 10 '18 at 15:47
  • "Believe me"... There is not a such closed form. Unfortunately I cannot prove this. Anyway, "believe me". – Ataulfo May 10 '18 at 16:03
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    time to define the Harris Number :) – tired May 10 '18 at 16:29

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