Are sin and cos the only continuous and infinitely differentiable periodic functions we have? 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:


*

*Square wave

*Triangle wave

*Sawtooth wave


Are there any other well-known periodic functions?

"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.
