periodic functionstrigonometry
Sin and cos are everywhere continuous and infinitely differentiable. Those are nice properties to have. They come from the unit circle.
It seems there's no other periodic function that is also smooth and continuous. The only other even periodic functions (not smooth or continuous) I have seen are:
Are there any other well-known periodic functions?
Given a periodic function $f(x)$ with period $2\pi$ and $f(0)=0$, there is polar function $$g(\theta) = f(\theta)/\sin(\theta)$$ that defines the shape. Note that this does not give you the values at $\theta\in\{0,\pi\}$; this is where you have to take a limit. If $f(\pi)\ne 0$ that might be difficult...
Shapes are thus unique, but if you have more than one place where $f(x)=0$, you can shift $f$ horizontally to get a different shape.
I can't do the others in my head, but the square wave is $g(\theta)=\left|\csc{\theta}\right|$, two horizontal lines.
EDIT here's the one for a triangle wave of maximum displacement $\pi/2$. Not exactly sure what function the curves follow.
EDIT 2 The sawtooth one, going from $-\pi/2$ to $\pi/2$, looks just like the triangle wave but it's only the right lobe, which gets drawn twice.
The simplest infinitely differentiable non-trigonometric* function I can think of is $$f(x)=\sum_{n\in\mathbb Z} e^{-(x-n)^2}\tag{1}$$ Periodicity is clear; differentiability follows from the fact that every derivative of $e^{-x^2}$ is of the form $p(x)e^{-x^2}$ for some polynomial $p$, and the series $$\sum_{n\in\mathbb Z} |p(x-n)| e^{-(x-n)^2}$$ converges uniformly on every bounded interval.
The function (1) is sometimes called the periodized Gaussian, although it seems that the same term is used for the nondifferentiable functions obtained by taking a central piece of Gaussian curve and repeating it.
(*) Not-explicitly-trigonometric. As others said, there is always a trigonometric series lurking in background.
Best Answer
"Are there any other well-known periodic functions?"
In one sense, the answer is "no". Every reasonable periodic complex-valued function $f$ of a real variable can be represented as an infinite linear combination of sines and cosines with periods equal the period $\tau$ of $f$, or equal to $\tau/2$ or to $\tau/3$, etc. See Fourier series.
There are also doubly periodic functions of a complex variable, called elliptic functions. If one restricts one of these to the real axis, one can find a Fourier series, but one doesn't do such restrictions, as far as I know, in studying these functions. See Weierstrass's elliptic functions and Jacobi elliptic functions.