The Cauchy-Riemann conditions for a function $f$ are respected if:
$$\frac{\partial f}{\partial \bar z} = 0$$
But is it a partial or a total derivative actually ?
If it is partial, then I can safely say that if $f = f(z)$, then:
$$\frac{\partial f(z)}{\partial \bar z} = 0 \; \; \; \forall f$$
For instance, $f(z) = \exp(z).\exp(\frac{1}{z})$ would be analytic $\forall z \neq 0$, thus.