[Boromir88]

But why did the man who "desired dragons" not include any in his greatest work?

I think it's due to show the extinction of this race. If we look at some fantasy movies, Dragonheart is the quickest one that comes to mind, there is only a couple dragons. There is also that recent John Rhys-Davies movie (which escapes my mind right now) about hunting a group of dragons who attacked the town. Point is, in a lot of fantasy there are only a couple dragons around, thought of as a race that is hard to multiple, that have been limitted in numbers due to being slain.

When you look at Tolkien's dragons, again there isn't that many, Ancalagon, Glaurung, Smaug, Scatha (I think I'm missing a couple, but oh well). All of them slain. So, why didn't Tolkien add in some dragons here? I would conclude to the decision of the growth of the race of men. We know as men grew some of the ancient races began to decline, Elves, Dwarves, Ents, some of these long, and more "old" then men, all began to decline. Maybe, that's what Tolkien was trying to get with dragons, as the race of men grew, this old, ancient race declined, and now they only remain in tales, stories, inns. Wonder if Tolkien is trying to show us something, when we start growing, or industrializing, we start suffocating, or destroying long ancient races that have been around for millions of years?

The farther we get into the stuff of which legends are made, the less we hear about dragons.

Very true, again the "tales" of dragons begin to fade as we reach the kingdoms of men (Rohan and Gondor). Stories of dragons are told in a society that is fascinated by dragon fireworks, has a "Green Dragon Inn," It's not until the hobbits are back in closing chapters when the hobbits are heading home, in Rivendell, with Bilbo, Smaug is brought up. So there are very few references to Dragons, but none of them are in the chapters that deal with the kingdoms of men, they are only now faint memories in the Shire, and in some ancient places, like Rivendell.