Is there a distribution for use with generalized linear models that captures both heavy tails and "pointiness" near the mean? If I fit a regular linear mixed model to my data with lmer, I get a pattern of residuals that, at a glance, looks to me to deviate from Gaussian in two ways.  The residuals are obviously very heavy-tailed, and there are a lot of discussions on Cross Validated of how to address heavy-tailed distributions.  But, for example, a commonly used distribution to address heavy tails, the Student's t distribution, doesn't capture this histogram at all.  The residuals are both heavy tailed and pointy -- there are extreme outliers and a lot of tight clustering around the mean.  Is there a distribution I can use as a link function in a generalized linear model (or some other approach) that would capture the actual shape of these residuals?
Here's a density plot for the dependent variable:

Here's one for the residuals in a linear mixed model:

Here's the Q-Q plot for the residuals:

EDIT:
Here's a residual versus fitted plot, to show the heteroskedasticity:

ANOTHER EDIT:
Here are density plots of subsets of the residuals, split at fitted = 1.1, in an effort to show the two distributions of the residuals with roughly homogeneous variance:



*

*You can't interpret the shape of the residuals without checking the conditional mean and variance assumptions (e.g. by residuals vs fitted); if the model for the conditional mean was wrong or the residuals were heteroskedastic, you could see a residual pattern like that even though the errors were normal.


*Assuming that's all fine, there's not a GLM that will do it, but an L1 regression (least absolute deviations regression) model might work reasonably well for conditional distributions close to a Laplace (you might want to check that the logs of the bin counts decrease roughly linearly either side of the mode; it can sometimes be hard to judge directly from the histogram, but it looks reasonable).
For an identity link and constant variance function, L1 regression is easy to do in R with quantreg::rq (with tau at the default value). There's other possible packages, but that's the one I'd look at first.
