User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-5029411-20130809210243/@comment-5150073-20130810105632

There can be ordinals which are indescribable in some notations. For example, $$\Gamma_0$$ is the first ordinal which cannot be described in the fixed up-arrow notation (by the "fixed" I meant that we define $$\alpha\uparrow^n (\beta+1) = \alpha\uparrow^{n-1} (\alpha\uparrow^n \beta)+1$$ rather than usual rule with "+1" at the end, in order to work with transfinite ordinals properly).