After watching a graduate course in Quantum Physics 2, and after hearing the lecturer saying a lot of times Dirac Delta.

It had occurred to me, can you have such a thing as a Dirac Delta of a Dirac Delta, etc?

Something like: $\delta(\delta(x-x_0))$, and $\delta^{n}(x-x_0):=\delta^{n-1}(\delta(x-x_0))$?

Does it have applications in physics?

How would you rigorously define such a composition of Dirac Deltas?

Another thing, would such a thing converge and in which sense would it converge if we let $n\to \infty$?