User blog comment:O0112111/Is w^2 a fixed poit of w+a=a?/@comment-35470197-20190614121128/@comment-35470197-20190614125225

Simply because w^^w itself is very important in mathematics. For example, w^^w is used in proof theory. Of course, w^^^…^^w might be also useful as long as you fix the definition. (The definition of w^^^w depends on the context unlike w^^w.)