An answer to the question "What regression/estimation is not a MLE?", a simple and robust alternative to Least-Squares (LS) is reportedly Least-Absolute Deviation (LAD).
To quote a source:
"The least absolute deviations method (LAD) is one of the principal alternatives to the least-squares method when one seeks to estimate regression parameters. The goal of the LAD regression is to provide a robust estimator."
Interestingly, per a reference, to quote "The least absolute deviations estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution." Here is a link that discusses some interesting applications of the Laplace (like as a Bayesian prior, and for extreme events).
Historically, the LAD procedure was introduced 50 years before the least-squares method (1757) by Roger Joseph Boscovich, who employed it to reconcile incoherent measures relating to the shape of the earth.
An illustrative difference is in the very simple case of Y = Constant, where the LS returns the sample mean, while the LAD selects the sample median! So in contexts with one or two extreme values, which for whatever reason (like heteroscedasticity), that may arise, LS could display a major shift in the true slope estimate, especially when there is one very low and/or a high observation, as a noted weakness. Wikipedia on robust regression makes a supporting comment:
"In particular, least squares estimates for regression models are highly sensitive to outliers."
With respect to applications, this can be particularly important, for example, in chemistry-based data analysis to predict a so-called reaction's Rate Law (which is based on the slope estimate).