Different for pure mathematics sure, but is that of practical importance given how fast busy-beaver numbers grow?
On a 32-bit computer, there are 2^32 bytes of memory or 256^(2^32)=2^34359738368 possible states. A program is something that takes you from one memory state to another and you want to figure out if the state transitions form a cycle?
Simplified a bit, the n-th busy beaver number is the number of 1s which can be written out by an n-state Turing machine.
For 2-symbols and n-states, the sequence goes: 6, 21, 107, >= 47176870, > 7.4×10^36534, > 10^10^10^10^18705353 — and those inequalities are because the numbers themselves cannot be computed directly, you have to run the program to see what it does.
[1] https://en.wikipedia.org/wiki/Transcomputational_problem
Although I personally believe hypercomputers are not physically possible, I'm not aware of any general proof that they are physically impossible. I've seen papers that suggest physical models for hypercomputers. Notice that there is a hierarchy of hypercomputers of varying powers and some of them can solve the Halting Problem.
Do you have a link discussing physical models for hypercomputers?
https://oronshagrir.huji.ac.il/sites/default/files/oronshagr...
https://link.springer.com/article/10.1007/s11047-009-9114-3
https://sci-hub.do/10.1142/S0129626412400105
An overview is in A. Syropoulos: Hypercomputation: Computing Beyond the Church-Turing Barrier. Springer 2008, chapters 8-9.
Gotta love the subsection headline in the third paper mentioned above: "A CTC Based Relativistic Hypercomputer That Actually Works Without Problems." ;-)