Let $\{f_n\}$ be the sequence of functions defined by this recurrence relation
\begin{cases} f_0(x) & = & x \\ f_{n+1}(x) & = & \ln(f_n(x) + 1) \end{cases} .
Is it true that $f_n$ converges pointwise to $g(x)=0$ on $[0,\infty)$ ?
If not, to what function does $f_n$ converges pointwise?
Yes, for each $x\geq0$, the sequence $f_n(x)$ converges to $0$. The easiest way to see this is to draw the graphs for $y=x$ and $y=\ln(x+1)$, from where you see immediately that $x>\ln(x+1)>0$ whenever $x>0$. Hence for each $x>0$, the sequence $f_n(x)$ is strictly decreasing and therefore converges to some $c\geq0$. This $c$ must satisfy $\ln(1+c)=c$, since the maps $x\mapsto x$ and $x\mapsto \ln(x+1)$ are both continuous on $x\geq0$.