zlacker

Ok, I think I may have the ultimate answer to this, a sequence that grows optimally fast. It goes like this:

Imagine 2^(7n) copies of the judge of this competition. For every judge create a different bit sequence of length 7n. Decode it from ascii and ask if it is a valid entry of the competition. Take the largest valid entry of those imaginary competitions. That is my entry. In order for this procedure to be consistent, this entry must be longer than n characters. Hence it will end with some padding you may disregard: asdfasdfasdfasdfasdfasdfasdfasdfasdfasdf.

>>im3w1l+(OP)
If you're allowed to submit entries that are mutually recursive with the decision procedure of the judges, I'm afraid the competition becomes less well defined. For example, it's easy to create a paradoxical entry by adding 1 to your entry. Also see Berry's paradox: http://en.wikipedia.org/wiki/Berry_paradox

>>cousin+c2
Yeah, I agree that it becomes less well defined. The problem with Berry's paradox is that it refers to itself. I circumvented paradox, by only refering to shorter descriptions of the number, than the description I gave. If that strategy works really depends on what the judge thinks though. Assuming he accepts refering to his own potential judgements of shorter sequences but not longer or equally long, the solution would work and the rules would be paradox free.

>>im3w1l+eH
Yeah, that might avoid paradox at the cost of requiring a specific kind of judge. But then I think you need to fully specify the judge's reasoning, and that's a very difficult problem, it might be even AI-complete. I don't know of any computer program that would be able to judge the descriptions of current winning entries.

Maybe this quote from the googology wiki will set you on a more productive track:

Googologists generally avoid many of the common responses such as "infinity," "anything you can come up with plus 1," "the largest number that can be named in ten words," "the largest number imaginable," "a zillion," "a hundred billion trillion million googolplex" or other indefinite, infinite, ill-defined, or inelegant responses. Rather googologists are interested in defining definite numbers using efficient and far reaching structural schemes, and don't attempt to forestall the unending quest for larger numbers, but rather encourage it. So perhaps a more accurate description of the challenge is: "What is the largest number you can come up with using the simplest tools?"

http://googology.wikia.com/wiki/Googology#History