Generalized CLT for any operation When multiple random variables are summed up, their sum converges to a normal distribution. If the RVs are instead multiplied together, the product tends to a lognormal distribution. There are some lesser known theorems that describe the distribution of the $\min\{X_1,X_2,...,X_n\}$ or the $\max\{X_1,X_2,...,X_n\}$ as $n\rightarrow \infty$ (by Fisher–Tippett–Gnedenko).
My question is, is there a generalized version of the CLT that describes the asymptotic distribution for any function of random variables? As in, the distribution of $g(X_1,X_2,...,X_n)$ as n tends to infinity?

There can only be such a theorem if $g$ is well-behaved in some sense: in particular, it should only depend on the set of values provided, not the order.
Here is a big class of such functions, to which the central limit theorem can be applied directly: functions $g$ for which there is an invertible function $f$ where:
$$g(X_1,X_2,...,X_n) = f^{-1}(f(X_1)+\dotsm +f(X_n))$$
Your example of the lognormal distribution is such case, where $f$ is $\log$. Your example of $\max$ is the limit as $k \to \infty$ of $f(x)=x^k$, and similarly for $\min$ with $f(x)=x^{-k}$.
