Hypothesis:
Given a non constant differentiable periodic function $f(x),\ \mathbb{R}\to\mathbb{R}$, which has an explicit finite form involving only elementary functions. The expression for $f(x)$ must include trigonometric functions, or equivalently, complex exponents.
In other words, we can't construct a periodic function from only roots, real exponents, logarithms and polynomials.
Is there a simple proof or counter example for the above hypothesis?
Yes. You are right. But I would like to propose a nearly-counterexample (which of course does not count, but still).
If you count the Bernoulli polynomials of negative order, then:
$$B_{-1}(x)+B_{-1}(1-x)=\psi^{(1)}(x)+\psi^{(1)}(1-x)=\pi^2 \csc^2 \pi x$$
This is a periodic function. And elementary. The only problem is, the Bernoulli polynomials of negative order are not polynomials or even elementary functions. But like Bernoulli polynomials of positive order they are cases of Hurwitz Zeta function.