There's quite a lot of applications. For example, I am currently working on research project in the context of auditing. The decision to conduct an audit can be thought of in the context of hypothesis testing.
Many finanicial transactions (e.g. writing checks) follow (not so well known) Benford's Law. The distribution associated with Benford's Law is multinomial (in the simple case 9 categories each of which having a particular probability).
Under a null hypothesis the data is assumed to be non-fraudulent. The alternative hyposthesis is that the data is fraudulent.
The sampling distribution, under a true null, follows Benford's law.
Thus, we have a so-called "goodness of fit" test. The traditional tests (chi-square, Kolmogorov-Smirnov, and z-tests) are not very good in terms of testing this hypothesis because they are too sensitive when sample sizes are large...i.e. these tests will essentially reject the null every time when sample sizes are large.
I'm currently working on a statistic that is not sensitive to large sample sizes that address this concern in the context of auditing.