I feel unusually obtuse in the presence of two patient and kind tutors.
Quote by Aether
If the two ends of the string had been moving apart at the speed of light for 100Gyr, then the dots 14Gly away would only be receding at 0.14 times the speed of light, would they not?

No, I am assuming an infinite string with constant rate of expansion such that dots 14B LY apart are receding FTL (which models our universe).
Quote by chronon
However, if you don't have any acceleration then Hubble's constant will decrease over time, since it is equal to speed/distance and the speed is constant while the distance is increasing. Hence your estimate of the age of the universe, which is based on 1/(Hubble constant) will increase.

Almost got a flicker of comprehension from this.
But I do not understand the part about the speed being constant, since from a fixed point the perceived recession rate increases with distance. Maybe I am off by one derivative of speed somehow...
Let me put forth my (a)ssumptions and (c)onclusions for you to demolish:
1a  assume no acceleration, gravity, or cosmological constant
2a  assume an infinite and eternally old string with dots at regular intervals
3a  assume the string between each dot expands at a fixed and constant rate
3c  this means that dots recede from all other dots at geometrically increasing rates as a function of distance
4a  assume the rate of expansion is equal to today's rate of expansion of the universe
4c  mathematically, dots 14B LY apart recede FTL
 (aside) this gives observable universes of up to 156B LY depending on whose math you believe in other posts
5a  you arrive at this string and attempt to determine the age of it
5c  you look in either direction and can see or calculate that the dot at 14B LY must be moving away at light speed
5c  you figure that dot must have been at your position 14B years ago
5c  you determine the string is 14B years old
I am completely aware that I am missing something painfully obvious. I just don't know what!