Examples of analytic functions to motivate a first course in complex variables [Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an example that comes up in other areas of mathematics? Particularly if one can comment on its natural domain (either a subset of $\mathbb C$ or an extension, i.e. a Riemann surface)—perhaps with special mention of any pole structure, conjectural or not—without getting into too much details of that particular field. 
Number theory of course offers a whole slew of examples: $L$-functions, modular forms (not to forget elliptic functions), where they encapsulate a lot of very deep mathematics. Would be great to be more specific. Of course the Riemann zeta function would be high on everyone's list.
What about "special functions" that solve differential equations (on the complex domain)? I'd also love to see a layman's example of a Painlevé transcendent.
I seem to recall Weierstrass's nowhere-differentiable function was discovered as the boundary value of an analytic function. Anyone know what it was? What about the smooth but nowhere-analytic function?
ADDED later: Let me explain a bit why I thought of asking this here––and I do it with some reservation. I was trying to write up an explanatory note on analytic continuation, and found that one could get across the idea (if not the actual theorem) without having to develop the basic theory of complex variables (along the line of Cauchy). In particular, I managed to give a list of "classes" of functions with increasingly sophisticated natural domains.


*

*polynomials and entire functions

*rational functions and meromorphic (on $\mathbb C$)

*algebraic functions like $\sqrt z$, and functions like $\log z$

*modular forms, etc., on the upper half plane that can't be extended at all
For each I wanted to give some nontrivial examples, so the students would see a wide variety of functions before going into the general theory. ("To see" may be taken literally: https://en.wikipedia.org/wiki/Domain_coloring.) I'm afraid too often in the standard complex variable course, when we are speaking of holomorphic functions, the students could only think of polynomials or rational functions when it comes to counterexamples or "checking" theorems. One of the books that tried to remedy that is


*

*Stalker, Complex Analysis: Fundamentals of the Classical Theory of Functions
which denotes the first half on "special functions" before going into the general theory. (I'm not sure how it would work in an actual course.)
So, the purpose for this question is solicit help in expanding and/or enriching the above list with more examples, especially ones that "open up" a whole subject (similar to the MO question Fundamental Examples), in a way that would be accessible and beneficial to students. Not knowing much myself, I have the impression that many new examples have come up since the "classical" special functions.
I do agree with the comment that there are way too many examples of analytic functions useful in other areas, which makes it even more startling since an analytic function is completely determined by its restriction on a small neighborhood, or by a sequence of numbers (be it Taylor series coefficients, or Dirichlet, or Fourier)––a point that perhaps is obvious but may seem surprising in the standard introductory course. My impression is that we don't stress this point enough.
 A: 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).   
A: These are the Mittag-Leffler function, entire functions of completely regular growth, and the Blaschke product.
