we show that there is a countable structure m such that for any set x, x computes a copy o m if and only if x is not hyperarithmetic. this gives a strong generalisation of the slaman-wehner theorem to the hyperdegrees. on the other hand, the generalization of the slaman-wehner theorem to the degrees of constructibility is false. this is related to work of kalimullin and his co-authors on structures whose degree spectrum has full measure. we show, for example, that there are only countably many such structures. we also touch on the possible form of structures m as above. for example, they can be linear orderings, but not have uncountably categorical theories. many open questions remain. among them: can we similarly capture the nonarithmetic degrees? joint work with antonio montalban and ted slaman.