[I had a different conjectural formula here before, but it is inconsistent with what we have since been told about f(3). Note that some discussion in comments below pertains to that formula.]
It has become clear that the problem we have to solve is a combination of the "overt" problem (here are the numbers, work out how to extend the sequence) and a "covert" problem made up of cryptic hints in the question. But let's see what progress if any we can make on the overt problem given the newly revealed value of f(3). Presumably the covert hints are meant to guide us to some formula (maybe involving counting trees on the edge-graph of a tesseract, or something) and it seems like it will have to be a fairly simple one.
We have f(3..14) = 18,14,12,11,10,9,9,8,8,8,7,7. In particular, note f(3)=18, f(3^2)=9=18/2. So perhaps we have something of the general form f(x) ~= a/log(x) (where the ~= indicates that we might be taking floors, or there might be some kind of offset, or "log" might mean something more like bitcount, or who knows what). That would explain the problems on day 1, and perhaps "logistical" is a hint at logarithms too.
Well, this quickly yields another simple formula that agrees with the given information:
f(n) = floor(20/log(n)). [We get the same output if we replace 20 with any number from 19.9 to 20.5, but I don't see any particular reason to find any such number preferable to 20.]
natural logs, presumably made from natural hardwoods rather than mysterious alien artefacts.
I will be moderately surprised if this turns out not to be the intended formula -- but I am still some way from figuring out all the cryptic clues.
If this is right, the total up to and including day 25 will be
172 floors made from the finest natural logs.
OK, what about the "covert" puzzle? Our answer involves
logs and floors
but it seems like maybe there is more. Let's take a look at some features of the story that might possibly be part of the covert puzzle.
PanGalactic Timber Company
Curious name. Might just be a Douglas Adams reference for fun, but maybe there's something more.
an Alien Artifact [...] the ruins of New York City, caused by those so-called "Avengers."
Sconibulus has indicated that this is a reference to a Marvel movie. The artifact is also called an "infinity stone" and "the tesseract" (the latter normally means a 4-dimensional analogue of a cube). There isn't anything in my answer that particularly reflects any of that. (In particular, the single magic number in my formula isn't the number of edges in a tesseract or anything like that.)
traverse the universe
Slightly odd turn of phrase. Is the answer supposed to involve some algorithmic thing involving e.g. traversal of binary trees?
the Artifact in question may still unfortunately have some alien remains sullying it
That's a bit weird. Also, this bit of the question was edited for no very obvious reason. So there may be a clue here, but I'm entirely at a loss to figure out what sort of clue.
each harvested tree is fastidiously cleaned before being converted into the finest hardwood floors you can imagine.
Sconibulus has already indicated that "floors" was meant to be a clue that the floor function features in the answer (to which I have to say: really? I mean, it's not like floors enter the question in some unnatural way that actually indicates something cryptic going on). But "fastidiously cleaned" and "the finest ... you can imagine" are interestingly hyperbolic, and "converted" wouldn't be the usual word here. Does the intended answer involve converting something to something else? Does it involve imaginary numbers?
(Perhaps fastidiously cleaning alien remains = subtracting off the remainder after taking the floor, in which case we've found a way to use more of the cryptic hints but it doesn't actually change what we're doing.)
a new U.N. building to memorialize the attack
Mention of the UN might be unnecessary detail -- is there a hint here?
(of course using our marvelous planking)
"Planking"? Unusual word. Reference to Planck's constant somehow??
our first three days of production were lost to a bore weevil infestation.
I haven't heard the term "bore weevil" before (though there are "borer weevils"); is there some coded hint here?
More fortunately, production measures have been put into place,
Curiously vague; some suggestion of "production systems" in theoretical computer science?
we faced an unusual Artifact-related problem the first day, which was a logistical nightmare.
This may be intended to hint at (1) a singularity at n=1 and (2) logs.
My guess is that there are more cryptic hints I haven't identified or decoded correctly yet.