The answer is no. If there was, then for some $N$ we would have $\|g^N\|<1/2$, in order for $g^N$ to be within distance $1/2$ of the constant $0$ function. But then for all $n\geq N$ we have that $g^n=g^N\circ g^{n-N}$ is bounded by $1/2$. In particular $\{g^n\}$ cannot be dense in $A$.

Essentially the same argument shows that if some constant function is a limit point of the sequence $g^n$, then it is in fact the limit of this sequence.

The answer is no. If there was, then for some $N$ we would have $\|g^N\|\leq 1/2$, in order for $g^N$ to be within distance $1/2$ of the constant $0$ function. This means that the range of $g^N$ is contained in $[0,1/2]$. But then the same holds for $g^n$ for all $n\geq N$, since $g^n=g^N\circ g^{n-N}$. In particular $\{g^n\}$ cannot be dense in $A$.

Essentially the same argument shows that if some constant function is a limit point of the sequence $g^n$, then it is in fact the limit of this sequence.