Given any continuous increasing function $\Phi:[0,+\infty[ \to ]0,+\infty[$ such that $\lim_{t \to \infty} \log \Phi(t)/ \log t = +\infty$, we show that there are harmonic functions $H$ on $\mathbb{R}^N$ satisfying the inequality $|H(x)| \leq \Phi(||x||)$ for every $x \in \mathbb{R}^N$, which are universal with respect to translations. This answers positively a problem of D. H. Armitage (2005). The proof combines techniques of Dynamical Systems and Operator Theory, and it does not need any result from Harmonic Analysis.