Non-trigonometric Continuous Periodic Functions

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.

One insight of Fourier was that more-or-less every periodic function should be expressible as a Fourier series $a_o+\sum_{n\ge 1} (a_n \sin nx+b_n\cos nx)$. The quality of the convergence depends on the smoothness of the function in question, unsurprisingly. So, apart from technicalities, every periodic function is so-expressible.

At the same time, it is very interesting to "make" periodic functions by taking something like a Gaussian $e^{-\pi x^2}$ (as in @NormalHuman's answer) and "wind up" by summing translates, to force periodicity. With a reasonable function $f$ (for example, in the Schwartz class: infinitely differentiable and it and all derivatives are rapidly decreasing), there is "Poisson summation" $\sum_{n\in \mathbb Z} f(n) = \sum_{n\in \mathbb Z} \widehat{f}(n)$, where $\widehat{f}$ is Fourier transform. This well-known and easily Google-able relation follows from the general expressibility of periodic functions by Fourier series. Among other things, this gives an expression for Fourier coefficients of by-force-periodic functions: by basic change-of-variable properties of Fourier transform, $$ \sum_{n\in \mathbb Z} f(n+x) \;=\; \sum_{n\in\mathbb Z} \widehat{f}(n)\,e^{2\pi inx} $$ A charming identity, in my opinion. :)

Take $$f(\theta)=\theta^4-2\pi^2\theta^2$$ for $-\pi\le x\le\pi$, and extend periodically. This is not explicitly a trig function, though as pointed out in a comment, by using Fourier series it can be written as an infinite sum of trig functions.

Also notice that while the derivative of $f$ exists, its third derivative does not. If you want an example which is arbitrarily often differentiable you will need something more intricate.