Another simple example that shows that the ML Estimator is not always unique is the model $U(\theta, \theta +1)^n$.
If your sample is $(x_1, ..., x_n)$ the likelihood $f(x_1,...x_n|\theta)$ for this sample is 1 if $x_i \in [\theta, \theta +1] \forall i=1...n$ and $0$ otherwise.