It is shown an example of a meromorphic function F in the plane, for which the exceptional set in the logarithmic derivative Lemma, in fact occurs. This example generalises a previous construction of Hayman. The function shown is given by a series of the form \[ F(z)=\sum^{\infty}_{n=1}(z/r_ n)^{\lambda_ n}, \] where \(\{r_ n\}\) and \(\{\lambda_ n\}\) are rapidly increasing sequences and it is proved that the quotient T(r,F')/T(r,F) tends to infinity when r tends to infinity through a sequence of intervals \(| s_ n,s_ n+\delta_ n|\) where \(s_ n\sim r_ n\), whence by an standard argument one concludes that the union of these intervals must be exceptional. Estimating the size of these intervals it is proved that some conditions on the size of the exceptional set are the best that one can expect.