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 not use GLM
To solve a generalized linear model (GLM) one uses a iterative algorithm that computes an ordinary least squares problem while changing the weights and values of the observations (also in the case of mixed models one can use an iterative algorithm to approximate the solution, for example Wolfinger & O'connell 1993).
Your function for the conditional distribution of the errors looks like a Laplace distribution or something similar (when you plot it on a log scale it is more clear), and can not be used as a distribution in a GLM (it needs to be in the exponential family).
Alternative
However, you can still use an alternative iterative algorithm. If your error distribution is the Laplace distribution then estimation of the model relates to minimizing least absolute residuals. For minimizing least absolute residuals instead of least squares residuals, on can use iteratively reweighted least squares.
Maybe this is a silly idea (one would have to back this up with some references), but I imagine that you can extend the iterative algorithm by replacing ordinary least squares regression (the function lm) with the mixed effects regression (the functionlmer) that you want to perform.
If there are no software packages that do this already, then it should not be too difficult to create a function yourselve (here is an example for GLM).
In R, the package lqmm provides the combination of mixed effect model with quantile regression.
