Find all real to real functions for which $f(x+y) +y$ does not exceed $f(f(f(x)))$ for all real $x,y$
I don't feel like substituting values is that useful, except probably substituting $x=0$, which gives a really nice bound on $f(y)$, but other than that, I can't quite find anything else useful, at least not in my opinion.
Choosing $y:=f(f(x))-x$,we imply : $$f(f(x))\le x$$ Thus $$f(x+y)+y \le f(x)$$ Thus $$f(x+y)+y \le f( (x+y)-y) \le f(x+y)+y$$ Hence, $$f(x+y)+y=f(x)$$ Hence forth the conclusion.
Comment
It seems to me that if we impose that $y$ must be nonnegative in the very first hypothesis, we will have non linear solutions.