Sure.

In my old job I modeled the transmission of sexually transmitted diseases; one variable is "number of sex partners in last year" (or last month or whatever). Another is "number of sex acts in last year". Both of these are oddly shaped, with minimum = 0, a strong peak at 1 and a long right tail. I used quantile regression, which doesn't make assumptions about the distribution of the residuals.

There are some other cases where it is generally thought that the distribution is normal, but it is not (in real samples). IQ has too many people in both far ends and has a different shaped tail on the right and left (the maximum is 200 or so, but the minimum is about 40). Weight of adults also has too many in the right tail and is not fully symmetric, but its right tail is also long (unlike that for IQ).