What is the maximum likelihood/GLM version of least absolute deviations for robust linear regression? Robust linear regression from minimising the absolute deviationresults in a regression line of medians conditional on covariates, instead of means using the standard least squares methodology: Is minimizing squared error equivalent to minimizing absolute error? Why squared error is more popular than the latter?
It is useful and robust in the sense that it minimises the effect of outliers in the response variable on the fitted line. What is its analogue in the generalised linear model/maximum likelihood setting?
 A: There's no GLM (no natural exponential family model) that corresponds to L1 (Least absolute value) regression.*
Note that if you're doing MLE then a density of form $\frac{c}{\phi}\exp(-g(\frac{y-\mathbf{x}\beta}{\phi}))$ with have log-likelihood $-n\log(\phi)-\sum_i  g(\frac{y_i-\mathbf{x}_i\beta}{\phi})$. 
Now maximizing likelihood with respect to the parameters in $\beta$ would correspond to minimizing $\sum_i  g(\frac{y_i-\mathbf{x}_i\beta}{\phi})$.
So if you're trying to minimize $\frac{1}{\phi}\sum_i  |y_i-\mathbf{x}_i\beta|=\sum_i  |\frac{y_i-\mathbf{x}_i\beta}{\phi}| $... the form of $g$ and hence of the density of the errors that this will be ML for should be immediately obvious -- it's the Laplace.

It is useful and robust in the sense that it minimises the effect of outliers in the response variable on the fitted line

It provides no protection at all against influential  observations, and so it's not at all robust to influential outliers, as illustrated here.

I also don't think it's quite correct to say it minimizes the effect; (ignoring the above point about influential observations -- e.g. if we're just looking at location rather than regression) it bounds the effect very nicely but if you learn about influence functions and M-estimators you'll see that there are estimators with influence functions that redescend (which L1 estimators don't), and so there's estimators of location where outliers have even less effect than they  do on the median. 

* Leaving aside the scale$^\dagger$ for simplicity, you just can't write $\sum_i  |y_i-\mathbf{x}_i\beta|$ in the form $\sum_i -\eta(\beta)\cdot T(y_i) +A(\beta)-B(y_i)$ - the absolute value function doesn't break up like that.
$\dagger$ Actually if we only had the scale to estimate, that would be exponential family.
