I was playing around with the family of sequences $s(x)$ defined for $x > 0$ as $s(x)_0 = x$, $s(x)_{n+1} = \log(1+s(x)_n)$ and I noticed a strange behavior. Specifically, regardless of the choice of $x$, $s(x)_{10^n} \approx 2\cdot 10^{-n}$ as $n$ grows:
> let logs x = x : logs (log (1 + x))
> logs 1000 !! 1000
2.0066131337990703e-3
> logs 1000 !! 10000
2.0008131530487204e-4
> logs 1000 !! 100000
2.0000966401761077e-5
> logs 1000 !! 1000000
2.000011198587142e-6
Based on this observation, I am wondering whether $n s(x)_n \rightarrow 2$ for all $x > 1$ and if so, why.
