Natural boundary
A boundary curve past which a power series admits no analytic continuation whatsoever.
A natural boundary is a curve bounding a power series' disk of convergence across which the function cannot be analytically continued at all -- not just along one path, but by any means, at any point of the curve. A function with a natural boundary is genuinely confined to its original disk; unlike a removable obstruction such as an isolated singularity, there is no way to patch around it and reach farther.
Examples
The series \sum_{n=0}^\infty z^{2^n} has the unit circle as a natural boundary, so it cannot be continued past |z|=1 anywhere.
Contrast a natural boundary with the single isolated singularity z=1 of 1/(1-z), which continues perfectly well everywhere else on that same circle.