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.