Let $u^*=\frac12(3-\sqrt5)$. As already noted by others, for every fixed $x$, $(F_i(x))$ converges to a limit which depends on $F_0(x)$: the limit is $0$ if $F_0(x)< u^*$, $u^*$ if $F_0(x)=u^*$, and $1$ if $F_0(x)>u^*$.
From here, the asymptotics depends on the starting point $F_0$. The OP is interested in the case when $F_0(x)=x$ for every $x$ in $[0,1]$. In this case, $F_i\to G$ pointwise, where $G(x)$ is $0$ if $x, $u^*$ if $x=u^*$, and $1$ if $x>u^*$.
This function $G$ is not a CDF but there exists a CDF $H$ such that $F_i(x)\to H(x)$ for every $x$ such that $H$ is continuous at $x$, namely the function $H=\mathbf{1}_{[u^*,+\infty)}$. Since $H$ is the CDF of a random variable $X$, a well known theorem asserts that this is enough to guarantee that $X_i\to X$ in distribution. Since $0\le X_i\le 1$ almost surely for every $i$, $X_i\to X$ in every $L^p$ as well. Of course, $P(X=u^*)=1$ hence $X_i\to u^*$ in distribution and in every $L^p$.
Note that for other starting points $F_0$, the limit can be different. For example, if there exists $w$ such that $F_0$ is the cdf of a random variable $X_0$ such that $P(X_0=w)=1$, then $F_i=F_0$ for every $i$, hence $F_i\to F_0$ for any $w$.
Likewise, if there exists $v and $p$ in $(0,1)$ such that $F_0$ is the cdf of a random variable $X_0$ such that $P(X_0=v)=p$ and $P(X_0=w)=1-p$, then one of three asymptotics occurs:
- either $p>u^*$, and then $X_i\to v$,
- or $p, and then $X_i\to w$,
- or $p=u^*$, and then $F_i=F_0$ for every $i$, hence $F_i\to F_0$.
A pathwise representation of the transformation of interest may help to get some intuition about the asymptotics described above and is as follows. Assume that the random variables $X_i^{(n)}$ are i.i.d. with CDF $F_i$. Then $F_{i+1}$ is the CDF of the random variable $X_{i+1}$ defined by $ X_{i+1}=\max\{\min\{X_i^{(1)},X_i^{(2)}\},\min\{X_i^{(3)},X_i^{(4)}\}\}. $