I am looking for a single word that is either a noun (like 'liar') or an adjective (like 'talkative') for a person who always has to have the last word in a discussion.
Don't bother arguing with Phil, he will always try to one-up you.
However, this doesn't really embody "a person who always has to have the last word." There are several other words that having the last word might be a symptom of, like stubborn, know-it-all, or egocentric, but again, being a last-worder is but a symptom of these traits.
Perhaps, if this person always needed to get the last-word in under their breath, we could call them an ultimumbler, a combination of the Latin word ultimum and the English word mumble.
While I do like saying ultimumbler, it looks horrendous on my screen.
An outtalker. Some definitions for outtalk:
To talk more, longer, or louder than (someone)
Outdo or overcome in talking