I found some neat stuff in Remmert's *Classical topics in complex function theory*.

Fabry's gap theorem gives a way to construct many examples including some already mentioned. Stated for the unit disk, it says:

If $m_1,m_2,\ldots$ is a sequence of positive integers such that $\displaystyle{\lim_{n\to\infty}}\frac{m_n}{n}=\infty$ and if $\displaystyle{f(z)=\sum_{n=1}^\infty a_nz^{m_n}}$ has radius of convergence 1, then the unit disk is the domain of holomorphy of $f$.

For example, if $p_n$ is the $n^{th}$ prime, then $$f(z)=\sum_{n=1}^\infty \frac{z^{p_n}}{n^2}$$ converges uniformly on the closed disk and is therefore continuous. It is not analytically extendable to any larger set because it satisfies the hypotheses of Fabry's theorem.

An interesting result that yields many such functions in a nonconstructive way is a theorem of Fatou-Hurwitz-Pólya:

If $\displaystyle{f(z)=\sum_{n=0}^\infty a_n z^n}$ has radius of convergence 1, then the set of functions $$f_\epsilon(z)=\sum_{n=0}^\infty \epsilon_na_nz^n$$ for $\epsilon_n\in\{\pm1\}$ whose domain of holomorphy is the unit disk has cardinality $2^{\aleph_0}$.

Hausdorff showed further that if $\displaystyle{\lim_{n\to\infty} |a_n|^{1/n}}$ exists (and equals 1) then the set of such functions whose domain of holomorphy is *not* the unit disk is at most countable. This applies in particular to the function $\displaystyle{f(z)=\sum_{n=1}^\infty \frac{z^n}{n^2}}$, which therefore yields examples by changing the signs of the coefficients in all but countably many ways.

One more, this time an explicit example from Remmert: The series $$f(z)=1+2z+\sum_{n=1}^\infty\frac{z^{2^n}}{2^{n^2}}$$ is one-to-one and has real derivatives of all orders on the closed disk, and has the open disk as domain of holomorphy.

Reference: Remmert's *Classical topics in complex function theory*, pages 252-258. (Fatou-Hurwitz-Pólya is stated on a page without preview.)