So I was thinking about Jonathan Bowers' Oblivion series numbers, and I came up with a similar idea. It's almost certainly just as undefined as they are, but I just wanted to post the idea and see if it was even theoretically workable.
First you start with the Church-Kleene Ordinal. Then, using the function of said Ordinal in the Fast-Growing Hierarchy, you define the number f(CK)(10^100^100) (Function Googolplex of the Church-Kleene function). Call this number CK0. Then you define the simplest (meaning using the fewest different symbols) system of mathematics that has a proof-theoretic ordinal equal to the Church-Kleene Ordinal. Call this system SCK0.
Next you take the largest finite number that can be defined in SCK0 using no more than CK0 symbols. This number is called CK1, and the function used to define this number will be called the CK0 function.
Then, find a fundamental sequence for an ordinal with growth rate comparable to the CK0 function (I know, easier said than done, but this is just the basic idea). Call this the CK0 ordinal.
Then define the simplest system of mathematics with a proof-theoretic ordinal equal to the CK0 ordinal. Call this system SCK1. Then take the largest finite number that can be defined in SCK1 using no more than CK1 symbols (the function used here will be called the CK1 function).
Repeat this process until you reach the largest finite number that can be defined in SCK(CK(CK(CK...(CK10^100^100)...))) (with CK(Googolplex) iterations), using no more than CK(CK(CK(CK...(CK10^100^100)...)))) (with CK(Googolplex) iterations) symbols.
I call this number the Ordinary Number.