Which of his filmography do you feel is his best, and why?

Social Network is what I would consider his best work. When it comes to the technical aspects; direction, cinematography and screenplay are fantastic and the best he is ever done IMO. However im a bit partial torwards Se7en and Fight Club, which I would consider my personal favorites if for no other reason than nostalgia.
Benji Button, Zodiac and Gone Girl would follow these in that order.

I'm in a pretty similar boat as you, though I'd put Gone Girl and Dragon Tattoo in the upper middle.

Need to get around to watching Panic Room and rewatching The Game some time...

For me it's gotta be Se7en, it's just a masterclass in atmosphere, cinematography, and creating a sense of a world on the edge of the abyss.

Fight Club and Zodiac are also superb, with the latter being, I feel, severely overlooked and underrated.

I think The Social Network is his best film. It explores the connection between humans (and lack thereof) in a nuanced and powerful way. I think Jesse Eisenberg's performance is spot on, and Fincher's direction is pretty much perfect, both when it comes to atmosphere, pace, structure and editing. The soundtrack is great too.
I also really love Zodiac and Gone Girl.
I could never really appreciate the extreme self-indulgence of Fight Club, the movie isn't nearly as smart or deep as it presents itself as. The music video style editing doesn't help either. It is still a solid film, with some fun ideas and good performances. But it's just that, solid.
I have yet to see Panic Room, Mank and The Game, but the rest of his films I would rank like this:

1. The Social Network
2. Zodiac
3. Gone Girl
4. Seven
5. The Girl with the Dragon Tatoo
6. Fight Club
7. Alien 3
8. The Curious Case of Benjamin Button

Se7en is my favorite. Zodiac is a close second though.

So it seems like nobody saw Mank so have my take: Mank is probably the least interesting Fincher film conceptually, so while it is well executed, it fails to leave a particularly strong impression. As such, I'd have to put it above Alien 3, but below all the other Fincher movies I've seen.

And Social Network is peak Fincher for my money.

I haven't seen The Social Network yet but a lot of the posts here are making me really want to. It just didn't sound like it would be very interesting, but then I guess a good director can make any story compelling.

I'm only familiar with Fight Club, Gone Girl, Alien 3, and Se7en.
It's Gone Girl for me. Why? I just liked it better.

I liked Fight Club before, maybe even more than Gone Girl. I think what ruined Fight Club for me were annoying memers who thought they were being clever by obsessively repeating stuff from the film.

Finally got around to watching Social Network.

It was actually much better than I was expecting; I didn't think it was quite as great as Se7en or Zodiac but it's still definitely still one of his best; great performances, cinematography, direction, and writing, kinda reminded me of Wolf of Wall Street a bit in so much as capturing the frantic, depraved world of attaining multi-million dollar success.