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.