Is there any practical application for performing a double Fourier transform? ...or an inverse Fourier transform on a time-domain input? In mathematics you can take the double derivative, or double integral of a function. There are many cases where performing a double derivative models a practical real-world situation, like finding the acceleration of an object.

Since the Fourier transform takes a real or complex signal as an input, and produces a complex signal as an output, there is nothing stopping you from taking that output and applying the Fourier transform a second time... Are there any practical uses for doing this? Does it help to model some complex real-world situations?

With the same logic, nothing would stop you from taking the inverse Fourier transform of your original time-domain input signal... would this ever be useful? Why or why not?
No, taking the Fourier transform twice is equivalent to time inversion (or inversion of whatever dimension you're in). You just get $x(-t)$ times a constant which depends on the type of scaling you use for the Fourier transform.

The inverse Fourier transform applied to a time domain signal just gives the spectrum with frequency inversion. Have a look at this answer for more details.