User blog comment:King2218/FGH Things/@comment-24509095-20140305140305/@comment-5529393-20140305145252

There's a problem with the property for the third hierarchy ordinal though: $$\omega$$ doesn't have a last element. You can say that it is the smallest ordinal a such that

a = f[0][0]...[0][a](0)

for any finite number of brackets.