User blog comment:Wythagoras/Proof that TREE(n+1) divided by TREE(n) is strictly increasing/@comment-223.255.230.227-20141220074940/@comment-1605058-20141220090746

If you allow m to be any real number, then maximal m doesn't exist. If you only want integer m, then maximum exists and is equal to ceil(log_3 TREE(3))-1.

What do you mean with m having a limit?