"Computability" is defined in terms of series and algorithms — it there's an algorithm to generate a series of numbers, the series is (by definition) computable

"Computability" is defined in terms of series and algorithms — it there's an algorithm to generate a series of numbers, the series is (by definition) computable. The Busy Beaver series of numbers stand out because there's no possible algorithm to generate them, per the halting problem. Also due to the halting problem, they grow faster than any possible computable series. In other words, for any series S(n), at some point you're going to hit a k such that for all numbers m greater than or equal to k, BB(m) > S(m). So in that sense a "smallest incomputable number" is not well-defined. However it is reasonable to call them the simplest possible description (not algorithm, just a description) of an incomputable series of numbers. There are more complex descriptions of incomputable numbers, such as the Busy Beaver number defined with respect to a Turing machine that has a busy beaver oracle attached to it.