Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x)=f(x+y+f(y))$ for any $x,y\in\mathbb{R}$ and $f(0)=-1$.
I suspect that the only solution is $f(x)=-1$. It is easy to see that we can reformulate that $y+f(y)$ is a period of the function. If we would know that this is a surjective function or its image contains an interval we would be done . Unfortunately not much can be said about it.
Note: It is false without an extra assumption like continuity. See example below.
(not an answer, but the picture won't fit in a comment)
This will do as well.