One additional example of non-uniqueness of MLE estimator:

To estimate the location parameter $\mu$ of the Laplace distribution through ML, you need a value $\hat{\mu}$ such that
$$ \sum_{i=1}^n \frac{|x_i - \hat{\mu}|}{x_i - \hat{\mu}} = \sum_{i=1}^n \mathrm{sgn}\left(x_i - \hat{\mu}\right) =  0,$$

so $\hat{\mu}$ must be below (or above) exactly half of the $x$'s, which means $\hat{\mu}$ is *a* median of them.

Even though when $n$ is odd we usually take the mean of the two central observations (in ascending order) as the median, it isn't unique. This may be a problem for numerical algorithms, and they can yield inconsistent results or even not converge at all.