Dragons of the Third Age

[Mithadan]

Even if we rely solely upon "canon", Scatha and Smaug were not the last dragons. In FoTR, "Shadow of the Past", Gandalf discusses dragons in the present tense as if they existed at that time (after Smaug was slain).

It has been said that dragon-fire could melt and consume the Rings of Power, but there is not now any dragon left on earth in which the old fire is hot enough...

And elsewhere in the same chapter, gandalf says:

Seven the Dwarf-kings possessed, but three he has recovered and the others dragons have consumed.

The hoard of Erebor did not contain a ring; Thror escaped with it. Thus at least four rings were consumed by dragons other than Smaug.

It is nonetheless possible that only two dragons escaped the fall of Angband. They may have been male and female and may have subsequently mated. Or they may have come upon whatever animals Morgoth corrupted into dragons and mated with those lesser creatures. Or, of course, more than two dragons may have escaped from the Battle of Wrath.