Let $G$ be an infinite residually finite group, and let $p$ be a prime. What are some examples where the finite quotients of $G$ have no elements of order $p$? Equivalently, what are some examples where $G$ has no subgroup of index a multiple of $p$?
So far the examples I can give are only when $G$ is periodic and no element has order $p$. Are there other essentially different examples, for example, are there torsion-free examples?
Edit: $\mathbb{Z}_q$ for $q \neq p$ is such an example, as pointed out in an answer. So the next question is: how about finitely generated examples?
The $q$-adic integers $\mathbb{Z}_q$ ($q$ prime number $\neq p$) is a profinite group, hence residually finite, since it is the inverse limit of finite groups. It is torsion free, since it has characteristic $0$ as a ring.
The finite quotients being $q$-groups, they have no element of order $p$.