Well, in case of power series some criterions do exist. Roughly speaking, one can take the element
$$\sum\limits_{n=1}^{\infty}c_nz^n,\quad z < 1,\qquad\qquad (*)$$
and consider an analytic function $\phi$, such that $\phi(n)=c_n$ for every $n$. $(*)$ can be analytically extended onto some angular domain iff $\phi(z)$ has finite exponential growth.
Let $E\subset \mathbb C$ be a closed unbounded domain and let $H(E)$ denote the set of functions such that each of them is analytic in a neighborhood of $E$.
For a function $\phi\in H(E)$, the exponential type of $\phi$ on $E$ is defined as
$$\sigma_\phi(E)=\limsup\limits_{z\to\infty,\\ z\in E}\frac{\log^+|\phi(z)|}{|z|}.$$
The following result is due to LeRoy and Lindelöf.
Theorem 1. Let $\Pi=\{z\in\mathbb C|\ \Re z \geq 0\}$. Assume that $\phi\in H(\Pi)$ is of finite exponential type $\sigma<\pi$. Then the series
$$f(z)=\sum\limits_{n=1}^{\infty}\phi(n)z^n$$
can be analytically extended onto the angular domain $\{z\in\mathbb C|
\ |\arg z|>\sigma\}$.
The LeRoy-Lindelöf theorem gives only a sufficient condition. A criterion can be obtained if we relax a bit the condition that $\phi$ is of finite exponential type.
Let $\Omega=\{z\in\mathbb C|\ \Re z > 0\}$ be the interior of $\Pi$. An analytic function $\phi\in H(\Omega)$
is said to be of (finite) interior exponential type iff
$$\sigma_\phi^\Omega=\sup\limits_{\{\Delta\}}\sigma_\phi(\Delta)< \infty,$$
where $\{\Delta\}$ is the set of all closed angular domains such that $\Delta\subset \Omega\cup \{0\}.$
Theorem 2. The element
$$\sum\limits_{n=1}^{\infty}c_nz^n,\quad z < 1,$$
can be analytically extended onto the angular domain ${{{{}}{}}}\{z\in\mathbb C|
\ |\arg z|>\sigma\}$ for some $\sigma\in[0,\pi)$ iff there is a function $\phi\in H(\Pi)$ of interior exponential type less or equal to $\sigma$ such that $$c_n=\phi(n),\quad n=0,1,2,\dots.$$
You might be interested in this article.
Weierstrass's function is the real part of
$$\sum_{n=0}^\infty a^nz^{b^n},\quad |z|\leq 1,$$
where $b\geq 2$ is an integer, and $a<1$. It was studied by complex analysis
in G. H. Hardy, in a series of papers, for example in TAMS, 17, 301-325 (1916).
Interesting examples can be of two kinds: a) useful examples, I mean useful in other areas of science and mathematics, and b) counterexamples, specially constructed to disprove some conjectures, or to show that certain assumptions
cannot be removed from some theorems, like the Weierstrass's function.
I will address only useful functions below.
Useful examples of analytic functions are solutions of various functional equations, first of all those related to differential equations.
Whittaker-Watson, vol. II is a good source for special functions which were
studied by the end of 19th century (Gamma, hypergeometric family (incl. Bessel, Airy, Weber, classical orthogonal polynomials etc.), Matieu and Lame functions, elliptic and theta
and Riemann zeta.
To this collection solutions of Painleve and Heun equations were added in 20th century, but this material is by far too advanced and cannot be addressed in a general complex variables course.
There are interesting functions with non-trivial singularities which arise in
holomorphic dynamics as solutions of functional equations of Schroeder, Abel and Poincare.
These are more easily accessible, and some of them can be included in a general course. (This is to address "interesting natural domains" in your wish list. They were actually studied for the first time by Fatou because of their funny
natural domains). Poincare functions give the famous Fatou-Bieberbach domains.
Other class of examples with interesting natural domains are automorphic functions, it is not difficult to give some simple examples, related to Fuchsian ot Schottky groups.
There are some other interesting and useful solutions of functional equations, my favorite one is the deformed exponential which solves
$$f'(z)=f(az),\quad f(0)=1,$$
where $a$ is a complex parameter, $|a|\leq 1$, but properties of this function,
for non-real $a$
except the simplest ones are still a complete mystery. For $a\in(-1,1)$ it
is well understood, and can serve as an example in a course of complex variables. It is related to certain generating functions in problems of physics and graph theory.
Of multi-valued functions, of course algebraic ones and Abelian integrals are the most important examples.
Ahlfors's textbook on complex variables covers the "minimal set" of most important functions listed above (Gamma, hypergeometric, elliptic, modular function and zeta).
Best Answer
You have to state a specific problem to get a reasonable answer. The proof of the statements you cite is trivial. A function is called analytic at $x_0$ if in some neighborhood of $x_0$ it is represented by its Taylor series at $x_0$. Here is does not matter whether $x_0$ and the neighborhood are real or complex.
But if a power series at $x_0$ is convergent at ANY point other than $x_0$, then it is automatically convergent is some COMPLEX disk around $x_0$.
This proves both statements that you cite. Now you are probably interested in the radius of the disk (or polydisk) in which it converges. One way to say something about it is to look how large the derivarives of $f$ at $x_0$ are. But I guess this you do not now. Then you need some information on how large is the complex neighborhood to which $f$ extends. For this you have to know something about your function.
So be more specific: what information about $f$ is available, and what you want to prove.