Log modulus has no global harmonic conjugate on the punctured plane ($\log |z|$ on ${\mathbb{C}}^{\ast }$)

The function $\log |z|$ is harmonic on the punctured complex plane, but it is not globally the real part of a holomorphic function there. Such a function would have derivative $1/z$, whose integral around the unit circle is $2\pi i$, whereas the integral of a derivative around a closed curve is zero.