I think it all hinges on what's meant by "with three nines".
Are you allowed non-digit symbols? In that case, there's obviously no upper bound. Just e.g. keep adding !s after a 9 and the number gets bigger and bigger.
Are you only allowed the 9s, with the only possibly difference in meaning being the positioning? Then 9^9^9 (where the ^ is represented purely by positioning on paper) looks to be the best you can do. I don't believe things like up-arrow notation can be represented with just positioning.
Are you allowed non-digit symbols? In that case, there's obviously no upper bound. Just e.g. keep adding !s after a 9 and the number gets bigger and bigger.
Are you only allowed the 9s, with the only possibly difference in meaning being the positioning? Then 9^9^9 (where the ^ is represented purely by positioning on paper) looks to be the best you can do. I don't believe things like up-arrow notation can be represented with just positioning.