There is a paper of Breuer and Simon, "Natural Boundaries and Spectral Theory" (some slides here ). They give, among other things, the definition of "strong natural boundary". This concept relates to "right limits" of the sequence of Taylor coefficients and give a criterion based on parts of the integrability of the original function on the boundary.

There is a paper of Breuer and Simon, "Natural Boundaries and Spectral Theory" (some slides here ). They give, among other things, the definition of "strong natural boundary". This concept relates to "right limits" of the sequence of Taylor coefficients and give a criterion based on parts of the integrability of the original function on the boundary.

I should also mention that these techniques can lead to a notion of "analytic continuation" through the natural frontier for a natural class of series (Sauzin and Tiozzo, "Generalised continuation by means of right limits" ).