For any given clustering task of interest, there is no single value of K.

Clustering & unsupervised machine learning is as much about creating meaning and structure as it is about discovering or revealing it.

Take the case of biological taxonomy, what K will best segment the animal kingdom?

There is no true value of K. If your answer is for a child, maybe it’ 7 corresponding to what we’re taught in school - mammals, birds, reptiles, amphibians, fish, and invertebrates.

If your answer is for a zoologist, obviously this won’t do.

Every clustering task of interest is like this. And I say of interest because clustering things like digits in the classic MNIST dataset is better posed as a classification problem - the categories are defined analytically.