$ f\circ f\circ f(x) = f(3x) $
Ignoring trivial (constant) solutions, I am not sure what I can try as an initial guess.
Also, how does this generalise?
i.e. If $f^k(x) = f \circ f \circ f\circ ...(x)$ composed $k$ times, then what are the generalised solutions? (if any)
For the case $f^n(x) = f(rx)$ for $n\in \mathbb{N}$ and $r\in\mathbb{R}$, you can substitute $f(x) = cx$ with $c\neq 0$ to get $c^n x = crx$ which is true when $c^n-rc = c(c^{n-1}-r) = 0 \Rightarrow c = r^{\frac{1}{n-1}}$. If $r\in\mathbb{Q}$, then we can write $r = \frac{p}{q}$, where $p\in\mathbb{Z}$ and $q\in\mathbb{N}$, and $gcf(p,q)=1$. Then, $c \in \mathbb{Q}$ if and only if $p^{\frac{1}{n-1}} \in \mathbb{Z}$ and $q^{\frac{1}{n-1}}\in\mathbb{N}$, and $c\in\mathbb{Z}$ if and only if $p^{\frac{1}{n-1}} \in \mathbb{Z}$ and $q = 1$. This is just one possible solution, there may be others as well.
