There is a bit of talk about the busy beaver function. It was asked whether or not it is possible to make a function that grows faster than the normal busy beaver function.
One way to do this is to use an oracle that knows the answer to the halting problem. This program is also a busy beaver function, called the second order busy beaver function.
Has there been any effort to compute or at least find lower bounds to the first value of the second-order busy beaver function?
Extra info: https://googology.wikia.org/wiki/Busy_beaver_function