What makes elementary functions elementary? Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are?  As I'm learning calculus, I seem to focus most of my attention on trigonometric, logarithmic, exponential, and $n$th roots, and solving problems that have solutions which are elementary functions.  I've been curious why these functions are called elementary, as opposed to some other functions that turn up rather naturally in mathematics.  What is the reason that these functions take up most of our attention, and is there a reason that some additional functions are not included amongst the elementary functions?  In other words, what property or properties do these functions possess that separates them from non-elementary functions (if there is one)?
 A: Another point, sometimes left out of the discussion, is the setting.  Something like: A meromorphic function defined on a connected domain in the complex plane is called an elementary function if ...  
Then you can do variants.  Functions defined on an interval in the real line, perhaps omitting trig functions in your definition ...  An abstract extension of a differential field $F(x)$ ...
A: Elementary functions are the functions built from the three binary arithmetic operations, addition $x+y$, multiplication $x\cdot y$ and exponentiation $x\text{^}y$ and their respective inverse operations, subtraction, division, finding a root and finding a logarithm.
A: As Sivaram Ambikasaran mentioned the description on Wikipedia is fine. 
I believe the class of elementary functions, $E$, is commonly thought of as a construction of the form 


*

*All polynomials are in $E$

*The exponential and the logarithm function is in $E$

*The sine and cosine functions are in $E$. 

*$E$ is closed under addition, subtraction, multiplication, division and composition (finitely many operations of these).

*$E$ is the smallest set with the properties 1-4.
This applies to both real or complex valued functions. 

Edit 1: 
Some examples of functions that are elementary 


*

*$f(x)=1-x^2$

*$s(x) = \sqrt{x}$ (see addendum below)

*$g(x)=\arctan x$ (see addendum below)

*$U(x)=\sin\frac{1}{\log(1+x^2)}$

*$A(x)=|x| = \sqrt{x^2}$

*$_2F_1(1,1,2,x) = \log(1-x)$ (a Gauss Hypergeometric notation that ends up in a elementary function)


Some examples of functions that are not elementary 


*

*The Sine integral $\operatorname{si}(x)=\int_0^x\frac{\sin t}{t}dt$

*The Error function $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$

*The Cantor function  

*The characteristic function of an interval.



Edit 2: 
The nomenclature elementary function is of course used since the functions that are elementary can be deduced using finitely many applications of elementary operations on basic "high school functions".  

Edit 3: 
Convinced by comments - removed closed under inversion in step 5. Added log in step 2.  

Addendum (2013-02-28):
To see that the square root function $s(x)=\sqrt{x}$ is elementary using the above definition just note that 
$$ x\mapsto \frac{1}{2}\log x = \log x^{1/2}=\log\sqrt{x}$$
and hence $$s(x)=\sqrt{x}=\exp(\log\sqrt{x})$$ is elementary too.
Perhaps $\arctan$ is a bit harder to deduce from the other steps in the construction of $E$, first note that 
\begin{eqnarray}
\tan x=\frac{\sin x}{\cos x}= \frac{e^{ix}-e^{-ix}}{i(e^{ix}+e^{-ix})}=
-i\frac{e^{2ix}-1}{e^{2ix}+1}=\\
-i\frac{e^{2ix}+1 -2}{e^{2ix}+1} = -i\frac{e^{2ix}+1 }{e^{2ix}+1}-i\frac{-2}{e^{2ix}+1}=\\-i +i\frac{2}{e^{2ix}+1}
\end{eqnarray}
Hence if we solve for, $e^{2ix}$ in the above identity we get
\begin{eqnarray}
e^{2ix}=\frac{2}{1-i\tan x}-1= \frac{2-(1-i\tan x)}{1-i\tan x}= \frac{1+i\tan x}{1-i\tan x}
\end{eqnarray}
and taking the complex logarithm (more precisely the principal branch of the logarithm) we get $$x=\frac{1}{2i}\log\frac{1+i\tan x}{1-i\tan x}$$
That is $$t\mapsto \arctan t =\frac{1}{2i}\log\frac{1+it}{1-it}$$
belongs to $E$.
A: The original truly elementary functions are rational functions built up with the basic arithmetic operations.
Arctan and log are necessary and sufficient to integrate rational functions while sin/cos and exp are the solutions of the simplest differential equations and serve as a basis for the simplest interesting class of differential equations.
So, in a way, trigonometric functions, log and exp are the next thing found if one adds calculus to basic arithmetic.
But of course, it remains always slightly arbitrary since it is hard to justify the exclusion of elliptic functions, for example.
Yet, few people would argue that the elementary functions aren't simpler/satisfy simpler relations than elliptic functions in most respects.
A: This is a sort of conjectural answer.
I suspect the elementary functions are called that simply because they are so simple. This sounds a bit tautological, so let me go on. What classifies as an elementary function? I think everyone would agree that polynomials, the nth-root functions, logarithms, exponentials, sines, and cosines are all elementary functions. A couple cool things about these functions are that we completely understand their graphs, continuity and rates of growth, and all derivatives and antiderivatives of all orders more or less. And we've seen polynomials and nth-root functions since elementary school; exponentials, logs, sines and cosines since mid-secondary school. One might argue that we don't really look at exponentials or logs until we prep for calculus - that's a fair statement, perhaps.
What I'm trying to say is that we've seen these all before and get them. When I think of a non-standard function, the first that come to mind are either the logarithmic integral $li(x) = \int \frac{1}{\ln x}$ and the sine integral $si(x) = \int \frac{\sin x}{x}$. I don't know why, really. But I have to really think before I know what the sine integral looks like (the logarithmic integral happens to not be so bad mentally, so it goes).
But what about other functions. Do we call them elementary? Are rational functions elementary? I might argue that a rational function is more elementary than $\sec x$, I think, because secant is weird (in my opinion). And we don't understand arbitrary antiderivatives of secants so well. To admit a weakness, I can't currently think of the antiderivative at all. (Is is $\ln|\sec x + \tan x|$? Something like that - differentiate it and find out, I suppose).
What about compositions of elementary functions. Is $\cos \sin (x)$ elementary? I hope not. But products are. Why? Again, I would say that products are easy, compositions are weird. Compositions can change domains and ranges in not-immediately-intuitive ways. I suppose products could too, but those fit my intuition better.
To give credence, natural log is a stretch for an elementary function in my opinion. It's well studied, but not well-behaved. We do understand arbitrary antiderivatives of it, but it takes a bit of work. This is a theme with natural log, just like asking how multiplying by $\ln (x)$ or $1/\ln(x)$ affects convergence (it 'almost' never does).
This makes me think of general reciprocals. $\frac{1}{x}$ is elementary. Is $\frac{1}{\ln x}$? I sort of hope not there too - it's close to the logarithmic integral.
So in short, I would say that elementary functions are those that we all like and understand without thinking too hard. But I also don't think that the exact list of elementary functions is concrete. It would be interesting to know what mathematica or W|A considers elementary, because they both certainly have an inbuilt list.
A: There is an article an the January edition of the Notices entitled "Closed Forms: What They Are and Why We Care" (PDF).
A: I believe that a system of axioms for elementary functions is not only hard to build, but also almost useless. Actually, there functions that some teacher/professor would call elementary and other teacher/professors would definitely not. For instance, I strongly suggest that characteristic functions of finite unions of intervals should be elementary: their graphs are simply horizontal segments, which is rather elementary. Moreover, students grow up thinking that every function is actually a $C^\infty$ function, since they believe that only "elementary" functions exist! It is time to allow elementary functions to be (at least) discontinuous at a finite numer of points.
More conservative collegues found a useful recipe: elementary functions are
1) polynomials;
2) exponentials and logarithms;
3) goniometric functions (sin, cos, arcsin, arccos, and their sons/daughters like tan, arctan, cotan, ecc.).
A: The term elementary function is rather historical than mathematical. Elementary functions are those which were first studied in the 17th and 18th centuries, built up with the basic arithmetic operations:


*

*polynomials

*exponentials and logarithms

*trigonometric functions

