Interpretation
One must be careful to treat an arrow chain as a whole. Arrow chains do not describe the iterated application of a binary operator. Whereas chains of other infixed symbols (e.g. 3 + 4 + 5 + 6 + 7) can often be considered in fragments (e.g. (3 + 4) + 5 + (6 + 7)) without a change of meaning (see associativity), or at least can be evaluated step by step in a prescribed order, e.g. 234 from right to left, that is not so with Conway's arrow.
For example:
The fourth rule is the core: A chain of 3 or more elements ending with 2 or higher becomes a chain of the same length with a (usually vastly) increased penultimate element. But its ultimate element is decremented, eventually permitting the third rule to shorten the chain. After, to paraphrase Knuth, "much detail", the chain is reduced to two elements and the second rule terminates the recursion.
Read more about this topic: Conway Chained Arrow Notation