User blog comment:King2218/FGH Things/@comment-5029411-20140304234740

f[0][1](n) = f[f[...](n)](n) (n f's)

f[1][1](n) = f[0][1]_{f[0][1]_{...}(n)}(n) (n f's)

f[0][2](n) = f[f[...][1](n)][1](n) (n f's)

f[0][0][1](n) = f[0][f[0][...](n)](n) (n f's)

The third hierarchy ordinal is

$$f[0][0]...[0][1](n)$$ (n zeroes)

Next extension!

third hierarchy ordinal = f[f[0,1]^w](n)