Consider the following
$$\frac{d}{dx}(\tan^3(\sqrt{x-1}))$$
In order to solve this problem, I (based on inspection) listed each of the functions from outer most to inner most as follows (keeping in mind that each of the following functions change in proportion to changes in $x$):
$$y=f(u)=u^3$$ $$u=f(n)=\tan(n)$$ $$n=f(r)=\sqrt r$$ $$r=f(x)=x-1$$
From here, I adapted the chain rule using basic fraction rules to the following:
$$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dn}\cdot\frac{dn}{dr}\cdot\frac{dr}{dx}$$
Which equates to:
$$\frac{dy}{dx}=3(\tan(\sqrt{x-1}))^2\cdot\sec^2(\sqrt{x-1})\cdot\frac{1}{2\sqrt{x-1}}\cdot1$$
$$=\frac{3\tan^2(\sqrt{x-1})\sec^2(\sqrt{x-1})}{2\sqrt{x-1}}$$
which (I've already checked) is correct. The question is however, will this method work for every composite function with more than two functions? (when done correctly, at least)
First, I would not use the same symbol, f, for all of these functions. Instead write $f(u)= u^3$, $u= g(n)= tan(n)$, $n= h(r)= \sqrt{r}$, and $r= j(x)= x- 1$ so that your function is $f(g(h(j(x))))$.
To answer your question, yes, the "chain rule" is generally applicable to any composition of differentiable functions.