I know that mathematicians are trying to construct adequate models for $( \infty, n)$-categories. Although, it seems to be an interesting task, I would like to know some explicity examples where this theory can be helpful.

In fact, for me there is no doubt that $( \infty, 1)-$categories are really useful. For example, the work of Lurie/Toën/Vezzosi in Derived Algebraic Geometry, or the Cobordism hypothesis. I have also the idea that $( \infty, 2)-$categories are used in the new advances in Langlands.

However, I do not know any useful application of $(\infty, n)$-categories, for $n \geq 3$. I can presume that they provide the necessary formalism to fill the missing details in Lurie's proof of the Cobordism Hypothesis, but if this is so people do not seem to care too much.