Are there oscillating functions that don't reduce to trigonometric functions? I am wondering whether there exists a class of oscillating functions that are distinct from trigonometric functions.  The only oscillating functions I could think of are of the $e^{ix}$ and $(-1)^x$ varieties, but these are easily expressed as trigonometric functions (or sums thereof).  I'm looking for functions that cannot be expressed as finite sums of trigonometric functions (but functions that are not themselves finite sums either).
I'm not sure how best to phrase this, but I'm looking for non-trivial answers to this.  Its trivial to find a function that has the same value at two distinct x-values and tile it infinitely.  I'm looking for something that can be expressed in a simpler way than an infinite piece-wise function.
 A: What about a "saw" function, with graph something like:
   .     .     .
  . .   . .   . .
 .   . .   . .
.     .     .

and then extended in the obvious way? But of course this can be represented via 
infinite sums of trigonometric functions ... that is called Fourier theory
A formula for the above saw function might be
$$
    f(x) = \begin{cases} x,   & \text{if $0\le x \le 1$} \\
                         2-x, & \text{if $1 \le x \le 2$}  \end{cases}
$$
and then extended periodically.
A: The graph of $f(x) = x \pmod n$ for any integer $n$ is periodic. In case you are not familiar with modular arithmetic, $f(x)$ is the remainder of $x$ after division by $n$. As an example, here is $f(x) = x \pmod 5$, courtesy of WolframAlpha:

A: Let $f(x) = e^{-1/x(x-1)}$ for $0<x<1$ and $f(0)=f(1)=0$. Extend periodically to all of $\mathbb R$.
The result is $\mathcal C^\infty$ and (by construction) periodic, but it is not a finite sum of trigonometric functions. Such a sum must be analytic but our $f$ is not analytic at the integers.
A: Mostly what comes to mind is BESSEL FUNCTIONS, of which there are many kinds. Note that $$ f(x) = \frac{\sin x}{x}  $$ extends with $f(0) = 1$ to an function that oscillates but cannot be written as a sum of sines and cosines. 
If you are insisting on genuinely periodic functions, you are out of luck, as such will have Fourier series if piecewise continuous. 
On the other hand, here is one that has roots at constant intervals and constant amplitude, 
$$   g(x) = \left( 2 + \cos \left( \frac{8 x^2}{\pi} \right) \; \right) \; \sin x        $$ 
but has no Fourier series. Would I lie?
A: To extend Austin's answer: as I have noted here, one could use the sawtooth function $x\bmod 1$ to represent any periodic function on the real line, if you know the "repeating unit". To summarize the point of that answer: if you want your function to be a repeated version of the function $f(x)$ over the interval $[a,b]$, and $f(a)=f(b)$, then $f\left(x\bmod(b-a)\right)$ is the function you want. One can then choose an $f$ that involves no trigonometric functions whatsoever.
A: If you extend the (-1)^x , where x is an unknown function,to coin-flips, or random {-1,1}.The series, S=∑{-1,1} can be a non-symmetric random walk with zeros proportional to the geometric mean of #+1's and #-1's. There are 2^n of these "functions". It may be impossible to express them algebraically or as trigonometric functions. They could be a "function" if stored on a computer.( I use 1e6 digits of pi as a source.)
