Assumptions and definition of the problem:
I consider the complex function
$$
f(x) = sum_{n=0}^M a_n exp(i k_n x),
$$
where $M>2$ is a finite integer, $x$ is a realvalued number, $k_n$ is a set of mutually incommensurate frequencies with $k_{n+1}>k_n$ and $k_0=0$, $a_n>0$ is a set of realvalued weights with $a_n<a_0<1/2$ and
$$
sum_{n=0}^M a_n = 1.
$$
Numerical evidence:
Studying the function $f(x)$ in the complex plane, a host of numerical examples show that the unwrapped phase diverges to $infty$ for $xrightarrow infty$. Note that for all numerical studies I used commensurate frequencies $k_n$, yielding a periodic function $f(x)$. The figure below shows an example with a period of 30 (you find here an animation). In the example in the figure, $f(x)neq 0$ for all realvalued $x$. In other examples, whenever $f(x)$ reaches a zero, then the unwrapped phase is assumed to jump by $pi$.
Conjecture:
I am looking for a rigorous proof that the unwrapped phase does indeed diverge to $infty$ for $xrightarrow infty$. In other words, I would like to prove that the function $f(x)$ winds an infinite number of times around the origin anticlockwise. Note that whenever $f(x)$ reaches a zero, the socalled unwrapped phase is assumed to jump by $pi$, by definition.
Observations:

It is important that $a_0<1/2$. On the contrary, if we have $a_0>1/2$, the normalization condition on $a_n$ imposes that no winding occurs. In fact, the weights $(a_0,a_1,a_2,ldots)$ can be continuously transformed to $(1,0,0,ldots)$, which corresponds to the trivial function $f(x)=1$, without that $f(x)$ ever crosses the origin.

It is important that $k_n$ are mutually incommensurate. Otherwise, we can find counterexamples for which no winding occurs. An example is provided by $k_n = n$ and weights distributed according to a Poissonian distribution,
$$a_n=exp(lambda)frac{lambda^n}{n!}.$$
Additional remarks
I am interested in this problem for my research work in physics, because it is linked to the behavior observed in some experiments, where we probe the quantum wave function of an atom in a trap potential.