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 $f(x|Theta)$ for this sample is 1 if $x_i /in [/Theta, /Theta +1], i=1...n$ and $0$ otherwise.

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.