I read about busy beaver numbers and how they grow asymptotically larger than any computable function. Why is this so? Is it because of the busy beaver function's non-computability? If so, then do all non-computable functions grow asymptotically larger than computable ones?
Great answers below but I would like to explain in plainer english what I understand of them.
If there was a computable function f that grew faster than the busy beaver function, then this means that the busy beaver function is bounded by f. In other words, a turing machine would simply need to run for f(n) many steps to decide the halting problem. Since we know the halting problem is undecidable, our initial presupposition is wrong. Therefore, the busy beaver function grows faster than all computable functions.