Example of Probability Distribution Not in Exponential Family

[mynameisfunk]

Is there a good example of a probability distribution where the support set does not depend on the parameters and is still not a member of the exponential family?

 

[Office_Shredder]

I'm a bit confused because it seems like almost every family of probability distributions will satisfy this. For example Gaussian random variables have a mean and a standard deviation, and their support set is all of $\mathbb{R}.$ On the discrete case you have things like Bernoulli random variables with a parameter p that take value 1 with probability p, and -1 with probability 1-p.