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One of the first examples (historically) of nowhere differentiable continuous functions was given by $a_{2^n} = 2^n$ and $0$ otherwise. Taking tensor powers of this function you get very irregular functions of the kind you want. By very irregular here I mean nowhere differentiable (and so at least not in $\operatorname{Lip}_1$, but maybe you can get much more). In any case these Fourier series are called lacunary (à la Hadamard) and there should be a lot of literature about them.