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Ben Wieland showed me an argument that $B\mathbf{Z}_p$ is not hypercomplete; a sketch of his argument (which uses the Sullivan conjecture) is reproduced as Warning 7.2.2.31 in "Higher Topos Theory". And $\mathbf{Z}_p$ is about as tame as profinite groups get (the topos even has finite cohomological dimension).
Perhaps worth pointing out: the ring spectrum in question can be made E_2, since it can be described as the Thom spectrum of a 2-fold loop map Omega^2(BU(1)) -> Omega^2(BU) = Z x BU.
By nonunital E-infinity space, I mean an algebra over some operad O where O(n) is contractible for n > 0, and O(0) is empty. There is a proof (in the homotopy coherent context) in my book "Higher Algebra": see Corollary 5.2.3.12.
