Let the convolution closure of a family of distributions be all convolutions of finite sets of distributions in the family.
Is there a parametric exponential distribution family (see https://en.m.wikipedia.org/wiki/Exponential_family) whose convolution closure is dense in the space of all probability functions with the $L_1$ norm? (or some other norm?)
My initial thoughts are that such a family, if it exists, can't have only symmetric or log-concave functions, as convolution preserves these.