This article is not that good. I prefer the wiki pages about it (https://en.wikipedia.org/wiki/Tetration) or for a more general approach the best is wiki about hyperoperation (https://en.wikipedia.org/wiki/Hyperoperation). As for notation the square bracket notation is the simplest.

Story time: A while ago some other parent, trying to be smartass, asked the kids in one of those outside school activity (this was before COVID) "what is the highest number they can write using only 3 digits". Of course the kids, who barely understood multiplication and just learned in math the power of (a^b) operation, said "999". He said is "9^9^9 and started to explain to them how large that number is. After he was done, I said "you know, they are right, the highest number using only 3 digits is 999, but you used special notation. Now, if the rules say that we are allowed to use special notation then 9^9^9 is not the highest number, but 9[9]9 is. And then I had to explain to him what is that for the next 30 minutes. I lost him somewhere around pentation because he insisted how big that number is and I started to calculate it using previous base (power of -> tetration -> pentation -> etc). In the end I had to tell him, that using bracket notation his number is just 9[3]3, which is lower than 9[9]9.