Well, that's not quite true. The normal distribution maximizes entropy for a finite mean and variance. But it's not guaranteed that certain physical processes have well defined means and variances. A power law is the maximum entropy distribution with a well defined mean, but undefined variance, on the range (0, infinity). [1]

Of course in reality, at some point, some other physical process starts to dominate and cuts off the power law. this then turns it into a power law with a different slope or an exponential cutoff.

Edit: I forgot to add that a star's light profile on an image is actually approximately Gaussian. But it isn't exactly Gaussian --- I seem to remember that the core more closely resembles a Lorentzian. I am obviously not an observer, otherwise I would have thought of that immediately!

[1]: https://en.wikipedia.org/wiki/Maximum_entropy_probability_di...