 Natural boundary of a holomorphic function

A boundary of a domain of a holomorphic function is natural if the function has no analytic continuation through any of its points. For $f(z)={\sum }_{n\ge 1}{z}^{2^n}$, each root of unity of order a power of two is a singular boundary point: along $z=r\zeta$ all sufficiently late terms equal $r^{2^n}$ and their sum tends to infinity as $r\uparrow 1$. These roots are dense on the unit circle, so no boundary arc admits analytic continuation.