Suppose that functions $f(x),g(x),h(x)$, all $\mathbb R^+ \mapsto \mathbb R$ satisfy $$f\circ g\circ h=kx^r$$ with constants $k,r \in \mathbb R^+$. Additionally we know that $$g=1-bx^{-a}$$ for constants $a,b \in \mathbb R^+$.
I want to find and characterize a set of functions that satisfies the above. One set that works is: $$f(x)=\left(\frac{1}{1-x}\right)^\theta$$ $$h(x)=x$$
where $\theta>0$ is a constant. Is it possible to prove that this is set is unique (up to the multiplication by any a set of constants, e.g. $f(x)=c_3\left(\frac{c_0}{1-c_1x}\right)^{c_2\theta}$ etc. - i.e. I would like to find all functional forms that satisfy this, not specific functions).
If not, is it possible to say anything general about $f$ and $g$?
unfortunately it is not possible to say anything general about $f$ and $h$. Observe that $g$ is injective. To see that note, that it is continuous on $\mathbb R^+$ and its derivative is strictly negative there. The same goes for $\mathbb R^-$.
That means you can find a left-inverse, i.e. you can find a function $g' \colon \mathbb R \setminus \{ 0 \} \to \mathbb R$ with $g' \circ g \equiv \text{Id}$. Now if $h \colon \mathbb R \to \mathbb R^+$ is any injective function and we also denote its left-inverse by $h'$, then $f$ with $$f : = (x \mapsto k x^r ) \circ h' \circ g'$$satisfies your desired equation.