User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-25418284-20130325203802/@comment-5529393-20130326040830

The supremum would be \(\Phi(1,0)\). Unfortunately, it doesn't work because it skips most of the ordinals. The Turing machines are just regular Turing machines, so, while you can get \(\omega_1^{CK} + \alpha\) for \(\alpha < \omega_1^{CK}\), I don't see how to get \(\omega_1^{CK} * 2\).

One thing you can do is to define 3^m 5^n to be a notation for the ordinal defined by the nth rank-\(\mathcal{O}(m)\) Turing machine.