So in other words, you want to have a more intuitive understanding of how tetration, pentation etc. is developed?
I'm sure you probably find tetration (which we'll denote by a double up arrow, analogous to the single up arrow ^ used for exponentiation) to be easy to grasp, as a^^b is simply exponentiating b times (a power tower of a's that is b high, solved right to left, or at the top of the power tower then working downwards). The reason this is simple is because you already understand exponentiation. For a young student that only knows up to multiplication, it would be fairly simple for them to grasp the concept of exponentiation as a^b is the value a multiplied together b times, and similarly, for a student to learn multiplication, a*b is just a added together b times. Notice that with recursion, if you understand a certain level, then you can easily learn the next level.
So the next step is pentation. Well obviously by recursion, a^^^b would be the value of a tetrated b times. But what does this mean? It's not as easy now because tetration isn't something that we've been exposed to very much, but we have shown that it is equivalent to saying how high the power tower should be. So then a^^^b is saying that we calculate a^^a (a tetrated a times) which gives us a new number (unfathomably large for a>3) and then we calculate a^^(a^^a) which is a stack of a's in a power tower that is a^^a high. Repeat this process until we've used up b amount of a's.
Beyond this it gets even fuzzier. Intuitively understanding the next step gets farther out of reach, which is why we use mathematics. Recursive operations allow us to follow the pattern without having to know where we are exactly or where we're going to end up.