One additional example of non-uniqueness of MLE estimator:
When you need to estimate the location parameter $\mu$ of the Laplace distribution through ML, you need a value $\hat{\mu}$ such that: $$ \sum_{i=1}^n \frac{|x_i - \hat{\mu}|}{x_i - \hat{\mu}} = 0$$
That is, an estimate $\hat{\mu}$ such that the number of observations below and above $\hat{\mu}$ are equal.
Clearly for $n > 1$ the solution will not be unique, unless the the two central observations (in ascending order) are the same.
For simplicity, we choose as estimate $\hat{\mu} = \overset{\sim}{x}$ (the sample median), because it satisfies the condition.