  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 H\"older
$\mathrm{RH}_r^\mathcal{R}(\mathbb{T}^\omega)$ 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 \[ \bigcup_{p \in (1, \infty)}A_p^\mathcal{R}(\mathbb{T}^\omega) =
\bigcup_{r \in (1, \infty)} \mathrm{RH}_r^\mathcal{R}(\mathbb{T}^\omega), \]
the unboundedness result applies also to all reverse H\"older weights.
