This is OEIS Sequence A067798.
Nothing else seems to be known about it; at any rate OEIS gives no
references to the literature, only a link to
a list of further such $x$
from Giovanni Resta that extends it from the 43rd solution, $722015$,
to the 76th, $1103806594815$, which is just a a bit larger than $2^{40}$.
(Thanks to **Robert Israel** for noting this link in his comment.)

It seems very hard to do anything with the problem other than make a few
elementary observations (e.g. all $x>2$ will be odd), do some heuristics
(biases mod $3$, $4$, $5$, and other small moduli; expected asymptotics),
and use yet more computing power to find a handful of further solutions.

A067798from the OEIS, are the same that the solutions of $$x=\phi(\phi(x)+\phi(x+1)-1)+\phi(\phi(x)+\phi(x+1))-1.$$ I've tested it for the segment of integers $1\leq x\leq 10^6$. I have not studied subsequent compositions. $\endgroup$