This is a spinoff of this question
Defining
$$f_0(x) = x$$ $$f_n(x) = \log(f_{(n-1)} (x)) \space (\forall n>0)$$
and
$$a_0 = 1$$ $$a_{n+1} = (n+1)^{a_n} \space (\forall n>0)$$
How to calculate
$$\lim_{n \to \infty } f_n(a_n) $$
(an "experiment" here, but (beware) I think WolframAlpha is using an approximate representation in powers of 10)
Edit
A table with the first few values (made with aid of hypercalc, as per Gottfried's suggestion in comments)
$$\begin{array}{cc} n & f_n(a_n) \\ 0 & 1. \\ 1 & 0. \\ 2 & -0.366513 \\ 3 & -0.239279 \\ 4 & -0.0771089 \\ 5 & -0.06133128660211943 \\ 6 & -0.06133124230008346 \\ 7 & -0.06133124230008346 \\ 8 & -0.06133124230008346 \end{array}$$