What was Voyager's final crew count?

In-canon, the starting complement was 141 and the final complement was either 150 or 144 (or 143) depending on whether you consider "Timeless" or "Author, Author" to be the last reference on the subject; These figures were confirmed in the following episodes;

Zelador;

STADI: That's our ship. That's Voyager. Intrepid class. Sustainable cruise velocity of warp factor nine point nine seven five. Fifteen decks. Crew complement of one hundred and forty one. Bio-neural circuitry.

Intemporal;

LAFORGE : If you succeed, countless lives will be affected.

CHAKOTAY : We're here to save one hundred and fifty lives. Our crew.

Autor, Autor

NEELIX: In my hat I hold one hundred and forty six sequentially numbered isolinear chips, one for every member of the crew. Each chip entitles the bearer to three minutes of uninterrupted comm. time with his loved ones back in the Alpha quadrant. Good luck.

For the record, the crew count stated by Neelix in Author, Author seems to incluir the EMH. Since that episode there was one canon death (Carey) and one departure (Neelix) resulting in a final count of either 144 or 143 depending on whether you imagine the EMH is considered an official crew member by that point.

Per IMDB, there were a considerable number of canon references to crew numbers;

In season 2 episode 1 "The 37s", there are 152 crew. In season 3 episode 2 "Distant Origin" alien sensors pick up 147 lifesigns. In season 5 episode 4 "In the Flesh", the Doctor implies there are just 127 crew. In season 5 episode 13 "Gravity", there are 152 crew. In season 5 episode 15 "Dark Frontier" there are 143 lifesigns. In season 5 episode 22 "Someone to Watch Over Me" there are 146 crew. In season 7 episode 20 "Author, Author" there are still 146 crew members despite there being several deaths since the events of "Someone to Watch Over Me"

As to the second part of your question (does the figure in Author, Author make sense?) as you can see on the excellent Ex Astris Scientia website the crew count for the USS Voyager was notoriously inconsistent, even for a Star Trek series. It would rise despite crew members dying in previous episodes and fall without any justification. In the final two seasons, the number seems to simply be fixado em 146 independentemente of the events of the series.

Not only that, but due to the vagaries of TV extra casting, the number of crew members seen (and nomeado) on the show actually exceeds the crew complement numbers mentioned in the various episodes.

I know old topic, but I have to put on my suggestions here :-)

Temporada 1

• Caretaker: The crew starts with 141 people, 13 crewmembers dead + 36 added = 164 crew ("more than a dozen died" (Nightingale) / "One quarter of the crew is maquis"(Repression) / Maquis= 32, Tuvok+Paris+Kes+Neelix=4)
• State of Flux: Seska leaves / later dead = 163 crew
• Faces: Durst dead = 162
• The 37s: Janeway claims there are 152 men and women on the ship. this is simply wrong, she mixed it with 162

Temporada 2

• Alliances: 3 crewman (Bendera + 2) die of Kazon = 159 crew
• Meld: Darwin dead = 158 crew
• Investigation: Michael Jonas dead = 157
• Deadlock: Add Naomi = 158 crew
• Innocence: Bennet dead = 157 crew
• Basics part 1: 2 unidentified crew dead, as Kazon enter Bridge + guard in front of exploding door = 155 crew

Temporada 3

• Basics Part 2: Hogan + Suder + 1 unidentified dead = 152 crew
• The Swarm: 1 unidentified crewman dies = 151 crew
• Warlord: Kes kills Crewman Martin = 150 crew left
• Unity: Mary Kaplan dead = makes 149 crew
• Distant Origin: the Voth scan Voyager, detect 148 lifeforms out from 149 crew (Doctor is a hologram, they cannot detect him)
• Displaced: The same number here, 148 lifeforms in habitat (without Doctor)
• Latent image (from season 5): Ahni Jetal dies at Stardate 50979 (means end of Season 3) = 148 crew

Temporada 4

• Skorpion + The Gift: Seven of Nine added / Kes leaves, but still alive = still 148 crew
• Scientific Method: 1 crewman dies = 147 crew
• Ashes to Ashes (from season 6): Lyndsay Ballard get killed from the Hirogen at Stardate 51563 = 146 crew
• Killing Game: 1 dead = 145 crew
• Living Witness: Regarding the Doctor, 3 crewman get killed. Besides this episode plays many centuries in the future and the Doctor is just a backup-module, I can not count the three dead. In my counting they got revived by the real Doctor, so no difference at all = still 145 crew
• One: 1 crewman dies = 144 crew

Temporada 5

• Timeless: Here Chakotay or Harry mention a complete crew of 150 people, they just rounded it up
• Gravity: Logically Tuvok mentions a crew of 152, this is simply wrong and not logical. Poor authors of this episode
• Dark Frontier: The Borg scan 143 lifeforms on Voyager, they cannot detect the Doctor = still correct 144 crew
• Someone to Watch Over Me: Neelix tells the ambassador the crew are 146 people, he uses a 1 year old crew manifest from Season 4.
• Equinox: Jurot + 2 crewman are dead = 141 crew Betazoid Jurot is one of the casualties, because she won't be available anymore in "Dragon's Teeth" 5 crewman of the ship Equinox are added = 146 crew

Temporada 6

• Between Equinox II / Imperfection: As mentioned in "Imperfection", Crewman Lang dies somehow between Equinox II and Imperfection, not seen in an official episode. He did not die with one of the 3 dead in Equinox, because he is still seen after the three others are already dead New crew = 145 people
• Collective: Add the 4 Borg-Children = 149 crew

Temporada 7

• Imperfection: Azan, Mezoti and Rebi leaves again the Voyager, Icheb stays = 146 crew Seven mentions, that until this episode 38 crewmembers were killed. Lang is one of them as seen in the list. In my count also 38 crewman are killed until now.
• Author Author: Neelix has 146 numbers for calling Earth in his hands
• Friendship One: Joe Carey dies = 145 crew
• Homestead: Neelix leaves / not dead: 144 crew
• Endgame: Miral Paris is born = 145 crew

From 44 lost persons 5 left ship (Kes, Borg-Children and Neelix) = so 39 dead crew at all.

There are lot of hints, that you have to take care, to reach the final count of the crew:

• More than a dozen died as Voyager got pulled in Delta Quadrant (“Nightingale”). So at least 13 people died.
• One quarter of the crew is Maquis (“Repression”). So from 140-150 people there should be at least 32 Maquis. From origin 141 crew 13 crewman did die in “Caretaker”. That makes 128 starfleet at the end of “Caretaker”, and a quarter of this is 32 Maquis. Count former Maquis Tom Paris and Spy Tuvok to it, then you even have 34. So this calculation also fits.
• From the beginning till end you have to fit the crew count the different people mention. Janeway in “The 37s” with a crewcount of 152, what doesn’t fit, because she mix it with 162 / “Distant Origin” + “Displaced” with 148 crew (149 crew with Doctor) / 143 crew scanned from the Borg (144 with Doctor) / the 146 crew count Neelix mentions in “Author Author”
• Most important is the number of 38 killed crewmembers, which Seven of Nine says in “Imperfection”. After this just Joe Carey is dead, no one else. This was one of the reasons, why I didn’t count the 3 dead crew in “Living Witness”(they got revived).

So Voyager arrives earth with 145 people of crew, three children (Naomi, Miral Paris, Icheb) among them. 39 people got killed, 5 left the ship. At all 145+39+5 = 189 names you can find in the crew manifest.

144

This is making two assumptions. OK, three actually. The first is that the number Neelix gives in Autor, Autor é preciso:

NEELIX: In my hat I hold one hundred and forty six sequentially numbered isolinear chips, one for every member of the crew. Each chip entitles the bearer to three minutes of uninterrupted comm. time with his loved ones back in the Alpha quadrant. Good luck.

This seems like a good number to take as an accurate count, considering the care that was probably taken in setting up the communication queue. There was a question as to whether Neelix gets a chip. In the transcript he gives it to EMH/Doctor so that the EMH can call his publisher:

EMH: No, but a rewrite could take weeks. My publisher is expecting a final draft tomorrow. NEELIX: Maybe you should give him a call. (hands over his chip)

After the EMH has made his first call. So both EMH and Neelix are included in the 146 count.

And secondly that except for the following two departures, there are no off screen deaths/abandoning of ships. Which as Richard notes in his comment on his answer, actually seems likely to have occured considering the random and often unexplained fluctuation of the number (which makes the 150 equally likely, imho).

However with that assumption we hit 145 when Lt. Carey is killed in Friendship One

[Bridge]

PARIS [OC]: are you doing?

(sound of weapon firing)

JANEWAY: Tom?

EMH: Sickbay to Bridge.

[Sickbay]

EMH: They've killed Lieutenant Carey.

And then we hit 144 when Neelix departs in Herdade.

The third assumption, as noted in the comments, is that we count the EMH/Doctor as a crew member. As he is given some legal rights at the end of Author, Author which convey at least partial personhood:

ARBITRATOR: We're exploring new territory today so it is fitting that this hearing is being held at Pathfinder. The Doctor exhibits many of the traits we associate with a person. Intelligence, creativity, ambition, even fallibility, but are these traits real or is the Doctor merely programmed to simulate them? To be honest, I don't know. Eventually we will have to decide because the issue of holographic rights isn't going to go away, but at this time, I am not prepared to rule that the Doctor is a person under the law. However, it is obvious he is no ordinary hologram and while I can't say with certainty that he is a person I am willing to extend the legal definition of artist to include the Doctor. I therefore rule that he has the right to control his work and I'm ordering all copies of his holo-novels to be recalled immediately.

And that he clearly has a position on the ship, I would lean to counting him as a crew member. Also, Neelix considered him a crew member in his original count (which is why he got a chip in the first place).

There is one important monkey wrench in the above estimates that comes in the episode "Displaced" from Season 3 (E 24). Ironically, the episode is mentioned several times here but not properly vetted for content it seems.

As crew members disappear Janeway establishes two important pieces of a math equation that lead us directly to the proper crew complement:

1. crew members are disappearing at a rate 1 crew member every 9 minutes and 20 seconds, or every 560 seconds
2. all crew members will be gone in 18 hours

Using these estimates, and with the understanding that there are 64,800 seconds in 18 hours, simple math shows that there are 116 crew members (rounding up even though rounding down would have been more correct, since rounding up favors a higher complement)

I concede to you that Janeway didn't grab a calculator and provide us with hard and fast numbers, so I don't think we should assume these are exact.

While it's tenable that Janeway's estimate should be taken with a grain of salt it should also be believed that, as the ship's captain and someone who is razor sharp at astrophysics and math, it's more or less accurate within an acceptable margin of error.

Most estimates here put the crew count around 146 at the time of the airing of this episode. If that is the case it would be implausible that Janeway would be this incredibly far off in her estimate of time needed for all crew to be replaced, as the necessary time would be over 23 hours, a far cry from her stated estimate of 18.

However, a countervailing monkey wrench is introduced by Chakotey just a few minutes later after Janeway is taken. it is stated at some point that the replacements are 1 to 1, meaning for each Nyran that enters the ship, 1 crew member disappears. When asked for an estimate of crew members left he is told here are 40, and then Chakotay states that more than 100 Nyrans are on board. By this estimate there are at least 141 crew members.

When addressing another episode's probably accuracy, joshbirk stated: "This seems like a good number to take as an accurate count, considering the care that was probably taken in setting up the communication queue."

Originally I agreed with this theory, but having seen this grand contradiction inside of one episode , and that contradiction coming within 4 minutes of the initial estimates from Janeway, I don't believe this anymore.

In the end I think the answer is very simple: The writers never truly and clearly made a final decision as to how many crew members there were.Furthermore, they were clearly careless in their deployment of numbers relating to quantity of crew.

Rather than explain the differences in crew count, they simply assumed no one would be the wiser if they either A) had a hard and fast number that they then played fast and loose with or B) just threw out random numbers in the 115-150 range throughout the series.

I genuinely think this was of not much concern to the writers and they did not take the actual crew count near as serious as us fans have and did not properly anticipate the level of detail to which we would take this in our desire to better understand this part of the ST universe.