We study maximal operators related to bases on the infinite-dimensional torus \(\mathbb {T}^\omega \). For the normalized Haar measure dx on \(\mathbb {T}^\omega \) it is known that \(M^{{\mathcal {R}}_0}\), the maximal operator associated with the dyadic basis \({\mathcal {R}}_0\), is of weak type (1, 1), but \(M^{{\mathcal {R}}}\), the operator associated with the natural general basis \({\mathcal {R}}\), is not. We extend the latter result to all \(q \in [1,\infty )\). Then we find a wide class of intermediate bases \({\mathcal {R}}_0 \subset {\mathcal {R}}' \subset {\mathcal {R}}\), for which maximal functions have controlled, but sometimes very peculiar behavior. Precisely, for given \(q_0 \in [1, \infty )\) we construct \({\mathcal {R}}'\) such that \(M^{{\mathcal {R}}'}\) is of restricted weak type (q, q) if and only if q belongs to a predetermined range of the form \((q_0, \infty ]\) or \([q_0, \infty ]\). Finally, we study the weighted setting, considering the Muckenhoupt \(A_p^{\mathcal {R}}(\mathbb {T}^\omega )\) and reverse Hölder \(\mathrm {RH}_r^{\mathcal {R}}(\mathbb {T}^\omega )\) classes of weights associated with \({\mathcal {R}}\). For each \(p \in (1, \infty )\) and each \(w \in A_p^{\mathcal {R}}(\mathbb {T}^\omega )\) we obtain that \(M^{{\mathcal {R}}}\) is not bounded on \(L^q(w)\) in the whole range \(q \in [1,\infty )\). Since we are able to show that $$\begin{aligned} \bigcup _{p \in (1, \infty )}A_p^{\mathcal {R}}(\mathbb {T}^\omega ) = \bigcup _{r \in (1, \infty )} \mathrm {RH}_r^{\mathcal {R}}(\mathbb {T}^\omega ), \end{aligned}$$the unboundedness result applies also to all reverse Hölder weights.