Under certain regularity conditions the CLT does indeed guarantee that a properly normalized sum of random variables converges to a Gaussian limit. But even in classical problems those conditions aren't always met.
The "law of rare events" gives one example of where a sum of independent random variables converges to a non-Gaussian limit.
Suppose we have a random process where at each step $n$, we observe $n$ independent binary random variables $X_{1n}, X_{2n}, \dots, X_{nn}$ where $P(X_{nk} = 1) = p_{nk}$ and $P(X_{nk} = 0) = 1 - p_{nk}$ (so they are Bernoulli with success probability $p_{nk}$). It turns out that if $\sum_{k=1}^n p_{nk} \to \lambda \in (0,\infty)$ as $n\to\infty$ and $\max_{1\leq k \leq n} p_{nk} \to 0$ then $\sum_{k=1}^n X_{nk} \stackrel{\text d}\to \text{Pois}(\lambda)$. This means that if we have a collection of binary random variables where the probability that any one of them is 1 goes to zero, but the collection as a whole maintains a steady expected number of 1s, then the sum will have a Poisson limit, not a Gaussian limit. This is theorem 3.6.1 in Durrett's Probability: Theory and Examples, available here.
Beyond this, we aren't always interested in means. Suppose we have $X_1, \dots, X_n \stackrel{\text{iid}}\sim \text{Unif}(\theta, \theta+1)$ and we want to estimate $\theta$. It turns out that $X_{(1)} := \min_{1\leq k \leq n} X_k$ is about the best estimator we could consider if we're using squared loss (in a minimax sense). If we normalize $X_{(1)}$ by subtracting $\theta$ and dividing by $1/n$ we get a non-Gaussian limit:
$$
\begin{aligned}
P\left(\frac{X_{(1)} - \theta}{1/n} \leq t\right) &= 1 - P\left(X_{(1)} - \theta > t/n\right) \\
&= 1 - P\left(X_1 - \theta > t/n\right)^n \\
&= 1 - (1 - t/n)^n \\
&\to e^{-t}
\end{aligned}
$$
as $n\to\infty$ hence $n(X_{(1)} - \theta) \stackrel{\text d}\to \text{Exp}(1)$, i.e. we get an Exponential distribution as our limit rather than a Gaussian.
These are two very classical examples of tidy problems where we don't get a Gaussian limit. If we're doing "real world" modeling then all bets can be off. We may have deep dependence relationships that prevent the CLT from applying, like if our data have temporal or spatial correlations. Or maybe it's a non-stationary process so there isn't one "mean" for things to be Gaussian around. Or we might be predicting/forecasting or studying non-asymptotic problems where there is no sense of "converging" to a limit. The CLT and friends are great but there is a lot of behavior that they fail to describe.
