I usually get drawn into their posts also and didn’t realize they were low value; my IQ must not be high enough to differentiate. What would you recommend as alternative sources?
All this talk about exponentiation, tetration and pentation have their roots in Peano arithmetic, which for reasons of logical clarity, defines precisely one function -- the "successor function":
s(n) = n+1
Using just this function, Peano defines addition, multiplication and exponentiation with great clarity. Since Peano's time, people have been making adjustments, like including zero among the counting numbers, and by extending exponentiation into tetration, pentation and a few more exotic operations fully discussed in the linked article and elsewhere.
I personally would have liked to see a more complete exposition of the logical roots of the Busy Beaver challenge, and I think the missing parts would have made the article a better read, even for non-specialists. Maybe especially for those readers.
But Quanta articles are perfectly suited to their audience, people who may be inspired to look for more depth elsewhere.
Anyway, this article seems fine to me. The "exponentiation" comment seems like a bizarre misreading. The article is just trying to explain how big BB(6) is. Before that, it explains what BB(n) is. To think it's solely about exponentiation you have to skip that entire section.
The problem with this particular article is simple: busy beavers numbers aren't interesting because they're big. They don't break mathematics because of that; you can always say "+1" to get a larger number. There's also nothing particularly notable about Knuth's up-arrow notation, which is essentially a novelty that you're never gonna use. Instead, the numbers are interesting because they have fairly mind-blowing interactions with the theory of computability.