Let $f_0(x) = x$. For $x > 0$, define functions inductively by $f_{n+1}(x) = x^{f_n(x)}$. So $f_1(x) = x^x$, $f_2(x) = x^{x^x}$, etc. Note that $f_0(1) = f_0'(1) = 1$.
Statements
(25) As $x \rightarrow 0^+$, $f_1(x) \rightarrow 1$.\\
(26) As $x \rightarrow 0^+$, $f_2(x) \rightarrow 1$.\\
(27) $\int_0^1 f_3(x)\, dx = 1$.\\
(28) The derivative of $f_{123}$ at $x = 1$ is 1.