New answer:
Three formulas for the $n$th derivative of a parametrically-defined function appear in Section 3 of Todorov's paper "New Explicit Formulas for the $n$th Derivative of Composite Functions," Pacific Journal of Mathematics 92 (1): 1981, pp. 217-236. Despite the English title, the paper is in German. None of the three formulas look like they would be particularly easy to use, but that's probably to be expected given the discussion in my original answer.
The first formula, apparently due to Pourchet, is
$$\frac{d^n y}{dx^n} = \sum_{k=1}^n B_{nk}(t) g^{(k)}(t),$$
where the functional coefficients $B_{nk}$ satisfy the recurrence relation
$$B_{n+1,k} = \frac{1}{f'} \left( B'_{nk} + B_{n,k-1} \right),$$
for $1 \leq k \leq n+1$ and $n \geq 1$. The boundary values are $B_{n0} = B_{n,n+1} = 0$, $B_{11} = 1/f'$.
The next two formulas are explicit and are due to Todorov, but they involve determinants. First we have
$$\frac{d^n y}{dx^n} = \frac{1}{(f')^{n(n+1)/2}} \begin{vmatrix} \dfrac{d}{dt} \dfrac{f}{1!} & \dfrac{d}{dt} \dfrac{f^2}{2!} & \cdots & \dfrac{d}{dt} \dfrac{f^{n-1}}{(n-1)!} & \dfrac{dg}{dt} \\
\dfrac{d^2}{dt^2} \dfrac{f}{1!} & \dfrac{d^2}{dt^2} \dfrac{f^2}{2!} & \cdots & \dfrac{d^2}{dt^2} \dfrac{f^{n-1}}{(n-1)!} & \dfrac{d^2g}{dt^2} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
\dfrac{d^n}{dt^n} \dfrac{f}{1!} & \dfrac{d^n}{dt^n} \dfrac{f^2}{2!} & \cdots & \dfrac{d^n}{dt^n} \dfrac{f^{n-1}}{(n-1)!} & \dfrac{d^ng}{dt^n}
\end{vmatrix},$$
which at least has nice symmetry.
The second general formula given by Todorov is
$$\frac{d^n y}{dx^n} = \frac{1}{(f')^{2n-1}} \begin{vmatrix} a_{11}(n) \dfrac{f'}{1!} & 0 & \cdots & 0 & \dfrac{g'}{0!} \\
a_{21}(n) \dfrac{f''}{2!} & a_{22}(n) \dfrac{f'}{1!} & \cdots & 0 & \dfrac{g''}{1!} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_{n-1,1}(n) \dfrac{f^{(n-1)}}{(n-1)!} & a_{n-1,2}(n) \dfrac{f^{(n-2)}}{(n-2)!} & \cdots & a_{n-1,n-1}(n) \dfrac{f'}{1!} & \dfrac{g^{(n-1)}}{(n-2)!} \\
a_{n1}(n) \dfrac{f^{(n)}}{n!} & a_{n2}(n) \dfrac{f^{(n-1)}}{(n-1)!} & \cdots & a_{n,n-1}(n) \dfrac{f''}{2!} & \dfrac{g^{(n)}}{(n-1)!}
\end{vmatrix},$$
where $a_{jk}(n) = (j-k+1)n - j$, for $1 \leq k \leq \min\{j,n-1\}$ and $1 \leq j \leq n$.
Todorov has some other formulas involving special cases of $f$, as well as results on the $n$th derivative of composite functions (as the title of the paper indicates).
In general, the operation on the right-hand side of $$\frac{d^n y}{dx^n} = \left(\frac{1}{f'} \frac{d}{dt}\right)^n g(t)$$ (which I have in my original answer) is called a Lie derivative. That appears to be the magic word here. (My running across this term in another setting is what led me to Todorov's paper.)
There are two special cases of this formula, given in Comtet's Advanced Combinatorics (p. 220), that are particularly interesting. If $f(t) = \log t$, then $1/f'(t) = t$, and $$\frac{d^n y}{dx^n} = \left(t \frac{d}{dt}\right)^n g(t) = \sum_{k=1}^n \left\{ n \atop k \right\} t^k g^{(k)}(t).$$
If $f(t) = \dfrac{e^{-at}}{-a}$, then $1/f'(t) = e^{at}$, and $$\frac{d^n y}{dx^n} = \left(e^{at} \frac{d}{dt}\right)^n g(t) = \sum_{k=1}^n \left[ n \atop k \right] a^{n-k} g^{(k)}(t).$$
**Original answer**:
In general, thanks to the chain rule, we have $$\frac{d^n y}{dx^n} = \left(\frac{1}{f'(t)} \frac{d}{dt}\right)^n g(t).$$
So a large part of the difficulty in finding a nice expression for $\frac{d^n y}{dx^n}$ lies in the form of $f'(t)$. The uglier $f'(t)$ is, the uglier this expression will be.
Looking at a simple case for $f'(t)$ should shed some light on the general case. Letting $f'(t) = 1$ is too simple, but the case $f'(t) = t$ (so, say, $f(t) = t^2/2$) gives some insight. See the end of the post for comments about what this case means for the general case.
If $f'(t) = t$ we have $$\frac{d^n y}{dx^n} = \frac{(-1)^{n-1}\phi_{n-1} (-t D(g'))}{t^{2n-1}} ,$$
where $D$ is the differentation operator (so $D^k(g') = g^{(k+1)}(t)$) and $\phi_n$ is the $n$th reverse Bessel polynomial.
The $n$th reverse Bessel polynomial is defined to be
$$\phi_n(t) = \sum_{k=0}^n \frac{(2n-k)! t^k}{(n-k)! k! 2^{n-k}}.$$
For example, for some small values of $n$ we have $$
\begin{align*}
\frac{dy}{dx} &= \frac{g'(t)}{t} = \frac{\phi_0 (-t D(g'))}{t}, \\
\frac{d^2y}{dx^2} &= \frac{g''(t)t - g'(t)}{t^3} = -\frac{ \phi_1 (-t D(g'))}{t^3}, \\
\frac{d^3y}{dx^3} &= \frac{g'''(t)t^2 - 3g''(t)t + 3g'(t)}{t^5} = \frac{\phi_2 (-t D(g')}{t^5}, \\
\end{align*}$$
which agrees with what J.M. has above. (Note that $D^0(g') = g'(t).)$ From here we do a proof by induction.
We will need $\frac{d\phi_{n-1} (-t D(g'))}{dt}$, so let's calculate that first. Once we get through the differentiation we'll substitute $z$ for $-t D(g')$, for simplicity's sake.
$$\begin{align*}
\frac{d\phi_{n-1} (-t D(g'))}{dt} &= -\sum_{k=1}^{n-1} \frac{(2n-2-k)! k (-t)^{k-1} D^k (g')}{(n-1-k)! k! 2^{n-1-k}} + \sum_{k=0}^{n-1} \frac{(2n-2-k)! (-t)^k D^{k+1} (g')}{(n-1-k)! k! 2^{n-1-k}} \\
&= \frac{1}{t} \left(\sum_{k=1}^{n-1} \frac{(2n-2-k)! z^k}{(n-1-k)! (k-1)! 2^{n-1-k}} - \sum_{k=0}^{n-1} \frac{(2n-2-k)! z^{k+1}}{(n-1-k)! k! 2^{n-1-k}}\right) \\
&= \frac{1}{t} \left(\sum_{k=0}^{n-2} \frac{(2n-3-k)! z^{k+1}}{(n-2-k)! k! 2^{n-2-k}} - \sum_{k=0}^{n-1} \frac{(2n-2-k)! z^{k+1}}{(n-1-k)! k! 2^{n-1-k}}\right) \\
&= \frac{1}{t} \left(- z^n + z \sum_{k=0}^{n-2} \frac{(2n-3-k)! z^k}{(n-1-k)! k! 2^{n-1-k}} \left(2(n-1-k) - (2n-2-k)\right) \right) \\
&= -\frac{1}{t} \left( z^n + z \sum_{k=0}^{n-2} \frac{(2n-3-k)! k z^k}{(n-1-k)! k! 2^{n-1-k}} \right) \\
&= -\frac{1}{t} \left( z^n + z^2 \sum_{k=0}^{n-3} \frac{(2n-4-k)! z^k}{(n-2-k)! k! 2^{n-2-k}} \right) \\
&= -\frac{z^2}{t} \phi_{n-2} (z).
\end{align*}
$$
Then we have, with $z = -t D(g')$,
$$\begin{align*}
\frac{d^{n+1} y}{dx^{n+1}} &= (-1)^{n-1} t^{-1} \left((1-2n) t^{-2n} \phi_{n-1}(z) - t^{1-2n} t^{-1} z^2 \phi_{n-2} (z)\right)\\
&= (-1)^n t^{-1-2n} ((2n-1) \phi_{n-1}(z) + z^2 \phi_{n-2} (z)) \\
& = \frac{(-1)^n \phi_n (z)}{t^{2(n+1)-1}} ,
\end{align*}$$
since the reverse Bessel polynomials satisfy the recurrence $\phi_n (z) = (2n-1)\phi_{n-1}(z) + z^2 \phi_{n-2}(z)$.
Since the Bessel polynomials show up even in this simple case, it appears that the general case will have to include some generalization of the Bessel polynomials. In fact, for general
$f'(t)$ the argument above goes through to show that the coefficients of the terms in the numerator of
$\frac{d^n y}{dx^n}$ containing only powers of
$f'(t)$ and
$f''(t)$ and derivatives of
$g(t)$ are the those of the reverse Bessel polynomials. What's remains, then, is to characterize the terms containing third and higher-order derivatives of
$f$.
