The only entries that take the lead on page 3 are Actaeus's entry, using iterations of Conway chained-arrow notation, and scikidus's last entry, using Bowers's planar array notation.
Blatm's entry is extremely large, but it is not presented in a computable manner, so it clearly should be disqualified under the OP's rules. Hialto's first entry doesn't halt. Hialto's second entry is troublesome because it depends on the halting behavious of Collatz sequences; it is theoretically possible that the particular sequence or sequences he chooses halt, but take a time longer than f_w^2(n) to halt, since I don't know whether there have been any upper bounds on the length of halting Collatz sequences proven. But, this seems unlikely in the extreme, and anyway the burden of proof is on him to prove that his number is larger. (Not that the others have, but in other cases we can clearly estimate their size.) Deedlit11 (talk) 03:05, June 27, 2018 (UTC)
Thank you! Adding page 3 as we speak. 2601:142:2:EC49:3444:C6CB:3B4E:33F7 12:16, June 27, 2018 (UTC)