It seems composition would yield ever larger numbers. If Rayo's number is F(10^100), then wouldn't F(F(10^100)) be even larger?
OK, now repeat that nesting F(10^100) times (maybe use Knuth's up-arrow notation?) and now you have something even bigger. Now call that whole thing G, and start over from the top.
There is no end to how many times you can do this.
In Googology, these kinds of operations are viewed almost the same as multiplying by a large constant. The result of everything you just did would be considered still in the neighborhood of Rayo's number, even if you did it rn^^^rn times. These people have a whole other notion of what "order of magnitude" means.
That violates rule #3: "Each new answer had to involve some new notion – it couldn’t be reachable in principle using methods that appeared in previous answers"
OK, now repeat that nesting F(10^100) times (maybe use Knuth's up-arrow notation?) and now you have something even bigger. Now call that whole thing G, and start over from the top.
There is no end to how many times you can do this.