In the central Limit Theorem it must be the case that the variance of $X_{i}$, where $\{X_{i}\}_{i \in \mathbb{N}}$ are i.i.d. random variables, must be finite. But can we find independent random variables $X_{i}$, not identically distributed with $\mathbb{E}(|X_{i}|) = \infty$ such that
\begin{equation} \frac{X_{1} + ... + X_{n}}{\sqrt{n}} \xrightarrow{d} \mathcal{N}(0,1), \end{equation}
where $\mathcal{N}(0,1)$ denotes a Standard normal variable.
Since we allow the $\left(X_i\right)_{i\geqslant 1}$ to be not necessarily identically distributed, the question is easier. Let $\left(N_i\right)_{i\geqslant 1}$ be an i.i.d. sequence of standard normally distributed random variables. It suffices to find an independent sequence $\left(Y_i\right)_{i\geqslant 1}$ independent of $\left(N_i\right)_{i\geqslant 1}$ such that $\mathbb E\left\lvert Y_1\right\rvert =+\infty$ and $n^{-1/2}\sum_{i=1}^nY_i\to 0$ in probability and define $X_i:=N_i+Y_i$. Take for example an independent sequence $\left(Y_i\right)_{i\geqslant 1}$ independent of $\left(N_i\right)_{i\geqslant 1}$ such that $Y_i$ follows a Cauchy distribution of parameter $c_i$ such that $n^{-1/2}\sum_{i=1}^n c_i\to 0$.