Busy beavers are a classic example. They're mostly-hypothetical numbers that tell you "if any Turing machine of size s runs for longer than this, it doesn't halt." There's a link to that in the sentence you quoted.
Since you asserted that individual BB(n) numbers are computable, I think you will have no difficulty writing an algorithm that outputs that.
This is a subtlety of doing computability theory in classical foundations. It’s akin to how every concrete instance P(x) of a decision problem P is decidable: just use excluded middle to figure out if P(x) is true or false, and then use the Turing machine that immediately accepts or rejects regardless of input. This is very different from writing a machine that has to decide P(x) when given x as an input!
I could write a few more given enough time, but writing later ones will take someone more omnipotent than me.
You may be confusing the true statement "for each n, BB(n) is computable" with the false statement "\n -> BB(n) is computable".