There are several recent results on non-trivial bounded functions $f$ that *are* orthogonal, i.e.
$$\sum_{n\leq x} \mu(n)f(n)=o(x),$$
to the Mobius function $\mu(n)$. These include the results of Green and Tao on Nilsequences (which includes information on the history of the problem) and Green on bounded depth circuits. The lecture notes by Peter Sarnak on Mobius randomness is an interesting overview of the subject.

Sarnak mentions there that it is easy to construct counter examples, or examples which decay very slowly. Of course, its easy to do so if $f$ is sparsely supported or has partial sums that are the inverse Mellin transform of a function having positive powers of $\zeta(s)$ as factors.

What are some examples of bounded, non-vanishing (or at least positive density) functions that

are notorthogonal to $\mu(n)$, and which don't come from $\zeta(s)$ in an obvious way?