User blog comment:Googleaarex/Unlimitable Ordinals/@comment-5029411-20130825150207/@comment-10429372-20130826125958

LittlePeng9 is right.

UL(4) = $$\varepsilon_{\omega^2}$$ Look at it like this: first fixed point of $$\varepsilon_{\alpha}=\varepsilon_{1+\alpha}$$ is $$\varepsilon_{\omega}$$, second is $$\varepsilon_{\omega2}$$, third is $$\varepsilon_{\omega3}$$, etc. So the limit is $$\varepsilon_{\omega^2}$$

UL(5) = $$\varepsilon_{\omega^{\omega}}$$ beacause $$\varepsilon_{\omega^2}$$ is the first fixed point of $$\varepsilon_{\omega\alpha}=\varepsilon_{\omega(1+\alpha)}$$, etc.

UL(6) = $$\varepsilon_{\omega^{\omega^2}}$$ UL(7) = $$\varepsilon_{\omega^{\omega^\omega}}$$ UL(8) = $$\varepsilon_{\omega^{\omega^{\omega^2}}}$$ etc.

UL_2(1) = UL(w) = $$\varepsilon_{\varepsilon_{0}}$$

From here, I'm not sure UL_3(1) = $$\varepsilon_{\varepsilon_{\varepsilon_{0}}}$$ UL_4(1) = $$\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_{0}}}}$$ UL_w(1) = $$\zeta_0$$