Why does the Wiener algebra not include all continuous functions: are there simple examples?

Question

If $(c_n) (n \in \mathbf Z)$ is a family of complex numbers indexed by the positive and negative integers that is summable (i.e. the sum of their absolute values is finite) then $f(x) = \sum_{n=-\infty}^\infty c_n \exp(inx)$ defines a continuous periodic function $f$ on the real line. All such functions build the Wiener algebra $A$. Has someone here a simple example showing that $A$ isn't the whole set of complex continuous periodic functions? In the theory of Fourier series there are examples with continuous functions having a Fourier series that doesn't converge everywhere; such functions might do the job, but they are always quite complicated. I am looking for simpler continuous functions not in $A$ if there is such a thing. Thanks