So a perfect fifth on B natural is F♯, an augmented 5th on B natural is F♯♯, would that then mean that an augmented 5th on B♯♯ would be F♯♯♯♯?
Technically yes, but you would almost never see B♯♯ as B♯♯ is an enharmonic equivalent to C♯ which makes much more senses in most cases.
Likewise I've never seen more than 3 accidentals applied to a note so a quadrupled sharped F you would never see. Going back to C♯, the equivalent interval would be G♯♯ or Gx better known as A.
So yes B♯♯ to F♯♯♯♯ is an augmented 5th, but you would most likely see it written as C♯ to Gx for everyone's sake.
Whilst technically there may be a key called B##, there would never be any reason for it. There's not even a good reason for the key of B#, which from a writing and reading perspective has no credibility over the enharmonic C. Here, we're talking of intervals rather than keys, but B## will almost never exist, so an augmented 5th on it would be rare too.
Sensibility would make a B## to be called C#,(or even Db) making the aug 5 Gx,(or even A) thus reasonably writable and readable for all concerned. Apart from that, there is no sign for triple or quadruple sharp, as there has been no perceptible need.