User blog comment:Edwin Shade/In The Pursuit Of Organization/@comment-1605058-20171030102151/@comment-1605058-20171030160435

In order to set \(a^{a^{a^{.^{.^{.}}}}}\) equal to \(x\), you have to know that it is equal to some real number. Consider an analogous situation with \(1-1+1-1+\dots\). Using your method, we could set it equal to \(x\) and then note \(x=1-1+1-1+\dots=1-(1-1+1-\dots)=1-x\), from which you would get \(x=\frac{1}{2}\). But the problem is that \(1-1+1-1+\dots\) is not equal to any real number: the sequence \(1,1-1,1-1+1,1-1+1-1,\dots\) is the sequence \(1,0,1,0,\dots\) which does not converge to any number.

If you think that this is a made-up example, I have to break it down to you that the same happens for \(a^{a^{a^{.^{.^{.}}}}}\) when \(a=0.05\), see here.