Simplest way to explain why CLT require independence and not uncorrelatedness? I am studying time series, and I am trying to understand why the CLT (central limit theorem) applies for independent observations, but not uncorrelated variables.  What is the easiest way to understand that uncorrelated variables is too weak?
 A: To restate the obvious, a CLT is describing the bulk error in a system comprised of numerous smaller components, under the assumption the errors of the smaller components can't somehow "conspire" together in a way that shifts the system as a whole. Independence is the simplest mathematical condition that prevents this kind of conspiracy from happening - although there are certainly others, which fall under the heading of weakly dependent CLT.
Uncorrelatedness does not prevent this sort of conspiracy. Let me try to give an intuitive explanation of what correlation means in this context. Let's say that the individual errors are centered (average out to zero). Then, pairwise correlations between the random components is measured by taking the average of the product. If it is zero, there is no correlation. But that does not prevent the random components from "conspiring" with one another - they just have to do it in a way that is symmetric around zero. (If you check out this counterexample linked in the comments, you will see a mathematical manifestation of what I mean.)
It might sound strange, but here is the analogy in my head. Imagine a prison as a random system, its individual components being the prisoners, and the CLT describing their collective sentiment. The warden wants the prisoners to be randomized with respect to one another, not conspiring among themselves - in other words, he wants the conditions for the CLT to apply. He enforces this by having the guards check that pairs of prisoners are not conspiring - effectively, it is a check regarding correlations. However, if a conspiracy is sufficiently subtle and no two prisoners know enough of the entire plan to expose it, then these correlations may all vanish - and yet the conspiracy persists. The conspiracy is a deviation from joint independence. And if a large enough group of prisoners collectively conspire on an escape plan, that is a deviation from the CLT.
Okay, now with that sufficiently wacky explanation out of the way, let me try to give an intuitive explanation of the counterexample I linked to. Think of it like a (bad) magic trick. The magician asks the mark to think of a random (but mean zero) number. Then, the magician asks the audience for a bunch of random numbers - let's say he does it in a way that tricks the audience into generating independent, mean zero random variables with unit variance. Then the magician asks the mark to multiply their hidden number by each of the audience's random numbers. And, voila: the magician furiously calculates and manages to guess the mark's number up to a sign - by calculating the standard deviation of the sequence of multiplied numbers and seeing how much larger (or smaller) it is than expected. Now as long as the mark's random number was not Gaussian, we will have contradicted the CLT - while maintaining uncorrelatedness between any two pairs of numbers in the sequence.
