What are special functions for?

If you read enough mathematics, you eventually come across several so-called "special functions". I'm always left wondering what on Earth these things are actually for.

We have the Euler Gamma function, the Beta function, the infamous Riemann zeta function, the Airy functions, the Möbius function, the Bessel functions, the Jacobi theta functions, the digamma and polygamma functions, the polylog, the Chebyshev polynomials, Weierstrass's elliptic functions, the Dedekind eta function, the hypergeometric functions, etc etc etc.

If you try looking any of these functions up, you generally get a huge page of dense, complex-looking formulas recounting the sixteen different ways to define the function, the myriad of properties and identities it has, and so on and so forth. But nowhere does it mention why this is useful.

I have just about figured out that the Euler Gamma function extends the factorial to non-integer values. I still don't know how that's "useful" in some way. Bessel J somehow appears at random in a few unrelated contexts, as do the Chebyshev polynomials. And that's about as far as I've got.

Obviously, I'm not expecting anyone to give an exhaustive explanation of what every single special function ever invented actually does. I suppose that if you don't know what a function does, that means you probably don't need it yet. But I would like to have some basic intuition as to why all these millions of functions have been defined, and what people use them for.

For each of the functions you mention, the answer to "why it's useful" will be somewhat different. In general, if a function appears in many unrelated contexts, that's a sure sign that it's useful. Think of other mathematical structures that appear in many unrelated contexts: the natural numbers, the real numbers, the complex numbers, polynomials, trigonometric functions, etc.

My understanding is that in the olden days, the second half a complex analysis course - after the theory has been developed - was a detailed study of special functions. (At least this may have been the case in courses for physicists.) That doesn't seem to be so common anymore.

Some remarks about specific functions that you mention:

Gamma function: Just as it turned out to be very convenient to allow the $x$ in $a^x$ to take non-natural-number values, it is convenient to allow the $n$ in $n!$ to do so. To give but one small example of an occurrence of the Gamma function in the answer to a naturally occurring question, the volume of an $n$-dimensional ball is $$ V_n(r)=\frac{\pi^{n/2}r^n}{\Gamma\left(\frac{n}{2}+1\right)}. $$ This example may not be altogether convincing, because the Gamma function can be eliminated in favor of more elementary functions, but it does have the advantage of providing a uniform formula for odd and even $n$, which the more elementary formulas to not. This is just the tip of the iceberg as far as appearances of the Gamma function in mathematics and physics go.

Functions defined in order to compute integrals: If the $\ln$ function were not already known, it would have been necessary to invent it in order to carry out $\int\frac{1}{x}\,dx$. The inverse of the the $\ln$ function would have then led to the exponential function. Similarly, if the inverse trigonometric functions, $\arctan$, $\arcsin$ were not already known, it would have been necessary to invent them in order to carry out integrals such as $\int\frac{1}{\sqrt{1-x^2}}\,dx$, $\int\frac{1}{1+x^2}\,dx$. This would then have led to the creation of $\tan$ and $\sin$.

In the case of elliptic functions, this actually is what occurred. The computation of the arc length of an ellipse (which is of interest in the study of planetary motion) leads to an integral that cannot be expressed in terms of elementary functions. This led to the definition of "elliptic integrals", whose inverses are elliptic functions, to which the theta functions are closely related.

Elliptic functions have the remarkable property of double periodicity. That is, they are periodic in two directions in the complex plane. You can imagine how this might be useful in the study of planar lattices, for example. But more generally, any time you have a function whose domain of definition is the torus, you will find that elliptic functions come into play. Certain complex algebraic curves of degree three are topologically tori; elliptic functions can be used to parametrize positions on such curves, much as $\sin$ and $\cos$ parametrize positions on the circle.

Solutions to differential equations: One of the first classes of differential equations you come to whose solutions cannot be expressed in terms of elementary functions are second order linear differential equations whose coefficients are polynomials of low degree. Hence the Airy functions, Bessel functions, hypergeometric functions, and so on. The simpler the differential equation, the more often it will come up. The differential equations that lead to the above functions arise repeatedly in classical mechanics, electromagnetism, and quantum mechanics.

Orthogonal polynomials: Principles of linear algebra can be brought to bear on the study of spaces of functions. Just as it is useful to write down an orthonormal basis for $\mathbf{R}^n$, it is useful to write down a basis for certain spaces of functions. Depending on the domain and symmetries of the functions of interest, different orthogonal polynomials will come into play. Just as periodic functions may be expanded as Fourier series in $\sin$ and $\cos$, series with other symmetries may be expanded in terms of other types of functions. Often the coefficients of the lowest order terms in the expansion have clear physical meaning. You mention the Chebyshev polynomials. Another example are Zernike polynomials, whose domain of definition is the disk, of which an example is the pupil of the eye. In optometry, the coefficients of some of the low order polynomials relate to certain refractive errors in the eye. In physics and chemistry, the spherical harmonics are used to describe atomic orbitals. They are useful in many other problems with spherical symmetry, including celestial mechanics.

In many cases the identities which arise from the analysis of special functions have interpretations in the context of analytic combinatorics or combinatorial species -- the partition function and Euler's pentagonal number theorem spring to mind. I also recommend looking at Herbert Wilf's Generatingfunctionology and Baez's twf:190. As V.I. Arnold says "...that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum."

It may also be useful to check out analytic number theory, recently on m.se.

I think you should take a look at the Bateman Manuscript Project, which does a better job of explaining the origin of each than either A&S or the DLMF.

I think, the question is stated "in a wrong way". Or "wrong" might not be the best characterization. It comes from the perspective of someone who "comes from outside" and is confronted with a set of conventions for approaches to problems and specific ways to solve them. So the "ousider" comes to ask: "what are all that special functions for". I think one should look at it from the "inside" standpoint: what are we doing? What problems in modeling physical or mathematical processes relations do we formulate, describe and formalize mathematically? For instance, "what is the sum of the squared reciprocals"? Well, this might be a problem which occurs often, and is at the heart of many mathematical problems in our daily work. So let's give it a name, for instance call it Zeta-function.
Clearly, there are functions, which occur nearly everywhere, like the power, the root function, then the exponential function, or the totient. That often occuring, simply stated relations, functions, series, basic for most of our operations (and may be: even of our thinking, mentally modeling) may be called "elementary functions". And functions, which are needed in less general and only specific context or depth, and possibly need much explanation, may then be called "special functions" .

So it might be understandable, why I feel that it sounds strange to ask "what are special functions for", instead of "which functions do mathematics/mathematicians call 'special' and which do they call 'elementary'?"