I've found kind of the reverse of what I'm looking for. I've found lower bounds for the length of a cycle when the cycle's members have a certain minimum value. For example, they can be found on Wikipedia here: https://en.wikipedia.org/wiki/Collatz_conjecture#Cycle_length
I'm looking for the reverse: For a given cycle length, I want to get lower bounds for the members of the cycle. I'm specifically interested in the lower bound of the maximum value in the cycle.