Whilst there are no definitive quotes about the height of Barad-dur we can make some vague estimations.
As already stated about the height of Orthanc:
"Between them was a narrow space, and there upon a floor of polished stone, written with strange signs, a man might stand [u]five hundred feet[/u] above the plain. This was Orthanc, the citadel of Saruman, the name of which had (by design or chance) a twofold meaning; for in the Elvish speech orthanc signifies Mount Fang, but in the language of the Mark of old the Cunning Mind."
But here is a folloewing quote which gives a gigantic estimate of how much bigger Barad-dur was then Orthanc:
"But Saruman had slowly shaped it to his shifting purposes, and made it better, as he thought, being deceived-for all those arts and subtle devices, for which he forsook his former wisdom, and which fondly he imagined were his own, came but from Mordor; so that [u]what he made was naught, only a little copy, a child's model or a slave's flattery, of that vast fortress, armoury, prison, furnace of great power, Barad-dûr, the Dark Tower, which suffered no rival, and laughed at flattery, biding its time, secure in its pride and its immeasurable strength[/u]."
If Orthanc at 500 feet was a 'little copy' of Barad-dur (of course this doesn't always refer to height, it also refers to the scale of the other dimensions of Barad-dur) it does indeed give a very large scope to work with.
We can narrow down the height a little more:
"Thus men reached at last the High Court, and the Place of the Fountain before the feet of the White Tower: tall and shapely, fifty fathoms from its base to the pinnacle, where the banner of the Stewards floated a thousand feet above the plain." (refering to Minas Tirith)
"Then turning south again he beheld Minas Tirith. Far away it seemed, and beautiful: white-walled, many-towered, proud and fair upon its mountain-seat; its battlements glittered with steel, and its turrets were bright with many banners. Hope leaped in his heart. [u]But against Minas Tirith was set another fortress, greater and more strong[/u]. Thither, eastward, unwilling his eye was drawn. It passed the ruined bridges of Osgiliath, the grinning gates of Minas Morgul, and the haunted Mountains, and it looked upon Gorgoroth, the valley of terror in the Land of Mordor. Darkness lay there under the Sun. Fire glowed amid the smoke. Mount Doom was burning, and a great reek rising. Then at last his gaze was held: [u]wall upon wall, battlement upon battlement, black, immeasurably strong, mountain of iron, gate of steel, tower of adamant, he saw it: Barad-dûr, Fortress of Sauron[/u]. All hope left him."
This is a little more vague. But from the first quote which names Orthanc as a little copy and Minas tirith being described as considerably smaller than Barad dur I think so far we can estimate a height of at least 1800-2000 feet.
Lastly... Mt Doom. Orodruin stood about 4500 feet from pinnacle to base, which is clearly described in Karen-wynn fonstadd's 'the Atlas of Tolkien's middle-earth'. The 'tumbled shoulders' of the mountain stood some 3000 feet high and then the central cone, which Sammath Naur was built into nearer the lower half stood a further 1500 feet. This places the Doorway into the mountain at around 3200 feet high. Now the next part is complete supposition but it is possible, and quite a 'Sauronish' thing to do, but I would say he built that doorway level with his topmost chamber in barad-dur where he gazed forth from with his palantir. Otherwise he could have built this doorway lower down in the mountain (or built barad-dur taller to match it).
Therefore concluding I would say Barad-dur is at the very least 2000 feet high but if you take my last theory I would place its estimate at between 2800-3400 feet.
But if you merely wished to know this to see if its the biggest fortress in Tolkien's myth then I would say no.
Thangorodrim were three mountains actually built by the orcs from the delving of Angband. They were hollow and clearly had many inhabitible chambers within. I would class this as a 'Built structure' of sorts. Therefore if you count this then the biggest built structure in Tolkien's myth measured, giving a good estimation from 'The Atlas of Tolkien's middle-earth', approximately 35,000 feet high, about 6.5 miles (roughly) and I beleive taller than Mt Everest which of course was not made by hands. It was the tallest peak in Middle-earth, surpassed only by the highest tower Oillose (not sure of spelling) on Taniquetil where Manwe and Varda dwell, and possibly* Kalorme, the highest mountain in the Lands of Sun, far east of Middle-earth across the eastern sea.
The only other mountain towers to ever surpass these three were the Pillars of Light which were made by the Valar and "Varda filled the lamps and Manwë hallowed them, and the Valar set them upon high pillars, more lofty far than are any mountains of the later days." These lights of course were afterwards overthrown by Morgoth.