Is there an instance where an iid random variable does not converge to Normal.

Question

I am an undergraduate student working on a thesis about the Central Limit Theorem. I am actually trying to construct a proof of an application of the CLT using the CLT. I was wondering however if there is a case where a random variable may be independently and identically distributed but still does not converge to Normal distribution. It is my understanding that most iid random variables satisfy the Lindeberg condition, but are there any examples where this is not the case? Any help is appreciated, thank you.