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:



The first thing that comes to mind is the double exponential distribution. It looks like there's a nimble package that might help (rdocumentation.org/packages/nimble/).
I'm pretty sure you couldn't use that with the generalized linear model, though. For estimation (fitting the model), a linear model would probably work. For testing, you'd want to use something other than the p-values that come by default. For example, bootstrapping might work.
