I can't quite understand how the Large Number Championship would work. A few key parameters aren't specified by the description I have, such as what sorts of notations are allowed, and if the contestants are allowed to define new notations, what standards govern their introduction, but that's not what's confusing me, as far as I can tell.
My first thought was something using Knuth's up arrow notation, and I felt pretty happy about that. It felt good, albeit not professional caliber, until I realized it was actually just someone else's idea.
So, the problem is that the contestants know the ins and outs of not only Knuth's notation but Conway's much more powerful chained arrow notation and any other demoniacal inventions out there (non-computable functions!!!!). It's quite plausible that they'll have prepared something entirely new for the contest. If so, they'll have to have a good method of determining which of the numbers is larger on the spot, which I find a mind-blowing task.