It's not "relevant" in any sense, but I think many computer folks are fascinated by busy beaver numbers because it's fairly easy to explain what the numbers mean - it's just a Turing machine with no icky math.

That said, functionally, they're more or less a restatement of the halting problem, essentially "if the halting problem was a sequence of numbers". Also see Godel's incompleteness. A bit mind-blowing that you have sequences of integers you can't "know" (i.e., compute or verify), but that's about it. There's no chance of this being useful in the system of mathematics we have today.