I had finished my Sith Inquisitor storyline and reached level 50, and had exhausted all of the conversation options on all of my companions (or so I thought), but noticed that I still had not unlocked the third companion (Ashara) in the Legacy Pane. At the time, I had about 7800 affection with the companion and realized that I probably still needed to hit some affection threshold to unlock those final few conversations. To plan which methods I would use to gain the affection, I wanted to find out the numerical value of the affection for the final conversation(s), but I could not find the number anywhere. So, I decided I would find out at least an approximate value for the sake of any others who may want to know.
To unlock the third companion's final conversation, you will need between 9278 and 9412 affection.
I determined this number by noting that I had 9277 affection and no conversation indicator before accepting the reward for a mission and gaining an the final conversation option after I accepted the reward. To gain the affection, I did the Belsavis dailies a few times over (three quests in particular will reward you 135 affection each: Unintended Consequences, Reluctant Volunteers, and Found in Translation).