As is usual, let's say an (_n_, _k_)-category is something with
objects, morphisms, 2-morphisms, ..., _n_-morphisms, such that all
_j_-morphisms for _j_ > _k_ are invertible, everything meant in the
weak sense. We can also take _n_ = ∞ or _n_ = _k_ = ∞. In this
terminology the weak ω-categories in the title question are
(∞,∞)-categories.

I think the only examples I know of weak ω-categories that are not (∞,
_k_)-categories for some finite _k_ are the ∞-category of all
∞-categories and the ∞-category `Cob` whose _n_-morphisms are
_n_-dimensional manifolds (with corners) thought of as cobordisms
between some specified (_n_-1)-dimensional manifolds (with corners).
(I saw Dominic Verity give a very nice talk about his construction of
a PL-version of this as a weak complicial set.) Of course, `Cob` has
many variants, and we could also look at constructions such as functor
categories, coproducts, products, etc., starting from these.

I'd be very interested in hearing about other examples of
(∞,∞)-categories, even if they haven't really been constructed in the
literature yet. Specially examples like `Cob` which are not internal to
the theory of (∞,∞)-categories.