A result of broader applicability is that of convergence to infinitely divisible distributions, more generally the stable distributions

https://en.wikipedia.org/wiki/Infinite_divisibility_(probabi...

https://en.wikipedia.org/wiki/Stable_distribution

This applies even when the variance is not finite.

Note independence and identical nature of distribution is not necessary for Central Limit Theorem to hold. It is a sufficient condition, not a necessary one, however, it does speed up the convergence a lot.

Gaussian distribution is a special case of the infinitely divisible distribution and is the most analytically tractable one in that family.

Whereas, averaging gives you Gaussian as long as the original distribution is somewhat benign, the MAX operator also has nice limiting properties. They converge to one of three forms of limiting distributions, Gumbel being one of them.

The general form of the limiting distributions when you take MAX of a sufficiently large sample are the extreme value distributions

https://en.wikipedia.org/wiki/Generalized_extreme_value_dist...

Very useful for studying record values -- severest floods, world records of 100m sprints, world records of maximum rainfall in a day etc