For my university degree studying foreign languages, I took a literature course covering fairy tales from all over the world, and the question on which I did my final dissertation for this course was on the significance of the number three in fairy tales. That being the case, I just wanted to add something in response to the question recently posed by Jake N. [click here for the original Q&A] on the significance of the number 3 in LOTR. These are my own ideas so feel free to disagree or completely rubbish anything I'm
I would suggest that the reason for there being so many instances of it's usage could stem from Tolkien's notion throughout the entire story, from The Hobbit all the way through to Sam's usage of it on his and Frodo's way to Mordor, of the third time paying for all. One could, I suppose, transpose many of the notions about the number 3 from fairy tales over to Lord Of The Rings, since it could indeed be seen as a fairy tale of sorts, if not quite as simplistic or straight-forward. In this response I would just like to give a quick summary of the kinds of ideas I came upon when researching my dissertation.
In these fairy tales, many of which, as you pointed out with your reference to the Bible being a source in which one finds many instances of the number 3 being of significance, one of the main characters is given trials or tasks to perform before he can be considered worthy or before he can pass a certain toll or boundary etc. He will inevitably fail the first two trials before succeeding the third time, since the first two are seen as things from which he must learn before being able to make the correct decision or follow the correct path. Thus, the number three eventually becomes considered as the path of wisdom and experience and the protagonist may only reach that path by treading the first two. In LOTR, Sam speaks of (and I'm paraphrasing here since I don't have the book in front of me unfortunately) "Third time pays for all, as my old Gaffer used to say". In saying this, he shows that the Gaffer may well have lived his life by this tenet and thus we can see him being, in comparison to the relatively innocent Sam, wise (in his own way). Bilbo, from the experience of his innocent beginnings at the start of The Hobbit to the experience and relative wisdom he displays in LOTR, also eventually speaks of the third time paying for all, and passes this notion onto to the younger Hobbits around him.
As you mentioned in your answer to the original question, three in LOTR also signifies the three Elven Rings and thus their bearers, who, in light of what I have said so far, are also seen to be very wise and experienced. One could possibly also suggest that the bearers of these three rings are in their own way divine. Indeed there is obviously a connection through the seeming immortality of Elrond and Galadriel and the manner in which Gandalf is resurrected after his battle with the Balrog. Thus, as in many instances in fairy tales from all over the world, the number 3 is related to divinity, to the Trinity, to completion and perfection. Things not related directly to the number 3 are equally considered of lesser worth, although there is often the notion that through the number 2 (be it of people, of events etc), the third can be attained. I don't want to go too deeply into that significance, save to say that it centres around the idea of the father, the mother and, eventually, the child (another very religious type of significance in terms of the number three), and thus onto the idea of progress, of new life, new hope etc.
In any case, this idea gives Elrond, Gandalf and Galadriel's overall roles in the story the feeling that they will triumph eventually over that which is lesser (The One Ring, the single Dark Lord etc). When Gandalf appears to be lost and seems to perish in Moria, there is a great feeling of absence, of foreboding and of an inevitable defeat, but upon his return, there is a feeling again of triumph and completion (or at least relief!).
Frodo and Sam (two) could not have managed to fulfil the quest on their own. They could get to the last hurdle but, without the presence of Gollum (in this case the third of their party), the task would have failed. I suppose that if you consider it this way, the number 3 does indeed become very significant in the story, if not necessarily in an immediately obvious way.
Ok, I won't bore you with any more of my rambling. There are many other instances throughout the tale of the number three cropping up, but it would take an entire Age of Hobbits, Dwarves, Elves and Men to discuss them all. Keep up the excellent work!
Gavin Anderson, Scotland