Earlier quoted context omitted.
If you somehow know BB numbers, you know the size of finite memory, which is sufficient to run corresponding busy-beaver. Limiting mathematics to what is physically possible is not useful. We don't know what is ultimately physically possible.
We have some upper bounds on what is physically possible: https://arxiv.org/abs/quant-ph/9908043
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.