We study the degrees of bi-hyperhyperimmune sets. Our main result characterizes these degrees as those that compute a function that is not dominated by any ∆02 function, and equivalently, those that compute a weak 2-generic. These characterizations imply that the collection of bi-hhi Turing degrees is closed upwards.

We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the set of all non-hyperarithmetic degrees. We also show that degree spectra can distinguish measure from category.