We will discuss some examples related to Newton's method. In particular, these examples answer questions by Buff and Douady. Let f be an entire function and let N(z)=z-f(z)/f'(z) be the associated Newton function. Douady had asked whether f must have 0 as an asymptotic value if N has an invariant Baker domain. Buff and Rückert showed that the answer is "yes" under suitable additional hypotheses. However, we show that the answer is "no" in general.

In the opposite direction to Douady's question, Buff and Rückert showed that N has an invariant Baker domain if f has a logarithmic singularity over 0. Buff asked whether N must have an invariant Baker domain if f has no zeros. We show that this is not the case. A modification of this example also answers a question of Rückert and Schleicher.