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.

EDIT: I think that Sam Gunningham is right and I forgot (again) that the difference between having duals and having inverses is supposed to fall of the edge of the world when you go all the way out to ∞, so that `Cob`

is an ∞-groupoid (specifically, it should be the well-known space classifying whatever kind of cobordism you used to build `Cob`

). This means that I actually don't know any examples of genuinely (∞,∞)-categories that come from outside higher category theory.

EDIT 2: I somehow missed this earlier question. Maybe my question should be closed as a duplicate.

EDIT 3: Jeremy Hahn has convinced me that Sam's comment is true or false depending on how you define the equivalences of (∞,∞)-categories, and that it is not clear whether you really want every (∞,∞)-category with all adjoints to be an ∞-groupoid.