Let A be a commutative ring and let O be a sheaf of E∞-ring spectra on SpecA such that π0O=OSpecA. Lurie provides a criterion when (SpecA,O) coincides with SpecO(SpecA), namely if the homotopy groups πnO are quasi-coherent sheaves on SpecA and O is hypercomplete (Spectral Algebraic Geometry, Proposition 18.104.22.168). To get a better understanding why this last condition is really necessary, I would like to know the answer to the following question:
Is there an example of sheaf O that is not hypercomplete, but satisfies the other conditions?
By Spectral Algebraic Geometry, Corollary 22.214.171.124, this cannot happen if SpecA is a noetherian space of finite Krull dimension. So a counterexample has to be something big like A=Z[x1,x2,x3,…].