Tolkien doesn't mention the size, but there are probably some ways to tell the size through reference points.

I'm sure there are. Tolkien was a stickler for details when he wanted to be.

"The Dark Tower was described as existing on a massive scale so large it was almost surreal, although Tolkien does not provide much detail beyond its size and immense strength. Since it had a "topmost tower" it presumably had multiple towers. It is otherwise described as dark and surrounded in shadow, so that it could not be clearly seen.
"..rising black, blacker and darker than the vast shades amid which it stood, the cruel pinnacles and iron crown of the topmost tower of Barad-dûr..." " - Wikipedia.

hmm, that would put Mt Doom on an even larger scale. The Atlas of Middle Earth places Barad-dûr as standing lower than the volcano, but still taller than half the height of the volcano.