Regression on a non-normal dependent variable I need to do a regression with a non-normal DV for which no proper non-linear transformation (that I know of) exists:

It is a score ranging from 10 to 50, with a high peak at 10, a drop at 11 and a regular decline from 11 to 50. The distribution of residuals is not normal.

There are more or less 500 subjects in the study, and the covariates are all dichotomous.
I thought of using n-tiles and performing an ordinal regression on the resulting variable, but then I end up with a high proportion of empty cells -- not because of a low sample size, but rather because of the several covariates that need to be in the model.
The best I could do for now is to remove covariates and use tertiles (instead of, say, quintiles) to minimize the number of empty cells. I am not sure, though, to what extent avoiding empty cells is THAT important. Compared to the original analysis on a raw score, regressing on tertiles with less covariates seems like a lot of sacrifices.
Based on the decent sample size, and given the fact that the distribution of residuals is not THAT far from normal, I am wondering if the results of the regression would be reliable as they are.
 A: Ordinal regression is not affected by empty cells of Y.  Quantile grouping is not required unless you just want to reduce computational burden.  Proportional odds or continuation ratio ordinal logistic models are likely to be able to handle the distribution of Y you plotted (with no grouping of Y).
A: The normality assumption is a convenient property of model's residuals, since it enables correct inferences about the estimated parameters and critical values of many other tests are also dependent on this assumption (therefore some corrections should be made, or you may roughly take more strict rule-of-thumb criteria, increasing the acceptable range of your tests), however it doesn't ruin the regression estimators. 
Thus it may (you still need to check the other assumptions) produce well behaved predictions, but data-mining and hypothesis testing would be a bit more difficult. At this point I do agree with Huber that you need to clarify the purpose of the model.

Regarding some tips:
At first glance it seems that your distribution after $Y-10$ transformation, could be approximated by some truncated versions of continuous distributions: exponential (Gamma), log-normal, Pareto or some other. So in log-normal case you still may move to something close to normality.
Another option could be to try something like fitting the combo of generalized logistic function and logistic regression. Since you DO know the upper and lower limits it seems feasible.   
