I'm not sure if it's bad form to suggest your own answer, but I've had a lot of fun writing one about the tallest structure built in Middle-earth, asked by iMerchant who has been contributing some pretty interesting questions about Tolkien, Star Trek and Star Wars.
The following is how I usually write any kind of answer that requires a lot of research.
Searching
People may think I know all of this stuff by heart, but I don't. I had some ideas on tall structures, but nothing really concrete. It was a combination of prior knowledge, PDF searching and googling ("middle earth tallest structure") that allowed me to come up with ideas.
There's a lot of information out there that can be used as a starting point, including the various Tolkien wikis and old-fashioned forums, but I have to personally verify everything I find. Some of the stuff I find turns out to be false, speculative or unverifiable. I either don't include it or add a warning that I couldn't confirm that particular piece of information.
I have all of Tolkien's main works, plus everything his son published in paper form, but I also have a lot of PDFs of various other documents. They're all in a folder and I use pdfgrep to search for certain terms in them. It's not always easy, especially since it doesn't handle non-ASCII characters very well, so anything with accents is kind of a pain to search for.
I once tried converting the PDFs to text files, but it didn't really help: all the formatting was lost (including table of contents, which are really useful) and non-ASCII characters were either lost, converted (é to e) or broken (é to e´), which made it even harder to search for. Although pdfgrep is slow and clunky, it's best I've found yet.
In some cases, I can either search for only part of a word if it's unambiguous (Oioloss for Oiolossë) or use regular expressions (F\w+nor for Fëanor).
Diving in
In his question, iMerchant mentioned "man/Númenórean/elf/dwarf/Maia/etc-built structures", which gave me the idea of splitting my answer by race. My initial idea was some of the taller mountains built by the Valar, such as Taniquetil, but he said geographical features did not apply.
My initial guess on the tallest structure overall was Barad-dûr, but I still started with the pillars first (even though they could be considered "mountains") because they are for me the largest project ever undertaken on Middle-earth.
Elves gave me some trouble, as they never really built anything fancy, as far as I could tell. I searched through all the PDFs I have for terms like "tower", "tallest" and "highest", but couldn't really find anything interesting. I also found various lists of buildings for more ideas, but all the suggestions in this part are speculative. Same thing for orcs. Dwarves were easy, especially since b_jonas had given the answer first.
I initially made a mistake for Men. I honestly didn't think of Minas Tirith and went for a tie between Orthanc and Armenelos first. It was only by going through the Atlas of Middle-earth for ideas that I saw Minas Tirith. That's why there's a weird "second place" subsection, where I kept my initial guess.
Then, while re-reading my answer on Dwarves, I thought of Melkor and Angband and integrated that in the "Valar" section.
I also had some issues with people editing my answer as I was writing it, creating some edit conflicts which, apparently, Stack Exchange is not very good at dealing with.
Touch ups and images
Once I was done with the meat of the answer, I did what I usually do with all my answers: add a proper introduction, some touches here and there (such as conversions into meters) and some links to various articles.
I also know people like having pictures, especially those from the Atlas, so I added some for Moria and Minas Tirith. Since I've bought the Atlas, I don't feel too bad about having a PDF and uploading low quality versions of the images.
This requires some work: taking a screenshot of the PDF, opening it in Paint.NET and cropping it correctly. Some images (like Thangorodrim) were split in two pages, so I had merge the images, but they weren't exactly aligned. I had to rotate them until they (mostly) looked good. If you look closely, you can see some repeating features that shouldn't be there and a pretty visible seam.
What I had to work with, the PDF of the Atlas in SumatraPDF
Additional information
I spend a lot of time re-reading my answers to find typos (ugh!), badly worded sentences or things I can add here and there. I usually find a lot of small things on the next day because I can look at the text with fresh eyes.
So I went back to Angband and Thangorodrim, because that part had been bugging me. I added more information on the height, including a drawing from Pictures, to make it clear that the estimate from Fonstad was based on some very flimsy evidence. The picture I initially included was a colored version, but I swapped it last minute with the original black and white. Although it doesn't look as good, it makes it clearer that there's not a lot of details.
Citations
Finally, once I was done with writing and formatting the whole thing, I went back to all my quotes and added citations for them. All the quotes I include in my answers come from various PDFs I have of Tolkien's works. They're easier to search and copy/paste, but they usually have weird page numbers (if any) that don't really correspond to anything. So I get my books out and search for the quotes in them. It's the part I hate the most when I write an answer, which is why I usually wait until the very end to do it.
Comments
While I was still working on the answer, people were starting to comment on it (I love it when that happens). I integrated information on the Temple's dome (from Paŭlo Ebermann) and clarifications on Moria (from Withywindle). iMerchant had a follow-up on the Paths of the Dead, but I didn't include it in the answer because it wasn't relevant.
Finally, Olivier Grégoire asked about the Hobbits. I had thought of it and did some quick searches at first, but I assumed (correctly, I think) that they wouldn't have built anything tall since they are uncomfortable with heights.
Phew
From my first comment on the question ("I have a feeling I'm going to spend way too much time on this question") until my last edit on the same day, I spent about 5 hours. That's typical of this kind of answer. It's a lot of work, but I love it. Keep those questions coming!
