Is there a function
F(x), such that for every x, larger than -1, the following is true:
F(x)=1+ F(x/(x+1)) ?
(In reply to Nearly correct answer (spoiler)
by Steve Herman)
I maintain that your 1st answer is the right one: there is no such function, and you've explained it loud and clear. X=0 prevents stating : "for all x ... "etc.
Nevertheless nothing's wrong with your additinal observations about "almost every x .." etc.