User blog:MilkyWay90/Yet another attempt at making a fast-growing function

I've really only made functions/notations that have a growth rate of only \(\omega ^ {3} + 1\), so I really, really hope that this goes higher than that.

RandomFunction(n) = The largest number that can be made in the language of Peano Arithmetic.

What is the growth rate of RandomFunction(n) and how can I improve it? I will also try extending it soon.