It depends what you mean by "structure"
In terms of freestanding structures, which I think you are most interested in, it would probably be Barad-dûr with a height of at least (and even perhaps much more than) 1000 feet (300 m), which would be about the same height as the Eiffel Tower. In second place would be a tie between Orthanc and the Temple of Armenelos on Númenor at about 500 feet (150 m). There are other tall towers, but their height is unknown.
For constructions that are above ground in general, Minas Tirith is the tallest at 1000 feet (300 m), but it was built from the side of a mountain. Underground structures are difficult to estimate, but it would probably be either Utumno or Angband, followed by Moria and its Endless Stair going to Durin's Tower.
Finally, if we go for "high mountainous things that were not really meant to be mountains", then the pillars made for the Lamps would probably take first place, followed by the whole of Angband and Thangorodrim together.
As a comparison, the Empire State Building stands at 1,454 feet (443 m), the Tokyo Skytree at 2,080 feet (634 m) and the Burj Khalifa at 2,722 feet (830 m).
I've looked for the tallest structure made by every race, although it's difficult to come up with even an approximate height for most of them.
Depending on your definition of "structure" and "mountain", the tallest thing the Valar built was the pillars on which the Lamps were set shortly after the creation of Arda:
Then 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. One lamp they raised near to the north of Middle-earth, and it was named Illuin; and the other was raised in the south, and it was named Ormal; and the light of the Lamps of the Valar flowed out over the Earth, so that all was lit as it were in a changeless day.
The Silmarillion, Of the Beginning of Days, p. 27
The height of the pillars is unknown, but it must be higher than "any mountains of the later days", which would include Thangorodrim. Karen Wynn Fonstad, in her Atlas of Middle-earth, estimates a height of 35,000 feet (10 km) for Thangorodrim, but it is rather doubtful (see below). If it is correct, then the pillars would have to be "far more lofty" than that. If we make a wild guess and double the height to 70,000 feet, or about 13 miles (21 km), the top of the pillars would be in the stratosphere.
It is not entirely clear what the pillars are made of. Fonstad assumes they are tall mountains, but I found no texts to support this. Indeed, in very early versions of the story, Melkor deceived the Valar and made them out of ice:
As is told they desired to make lamps, and Melko offered to devise a new substance of great strength and beauty to be their pillars. And he set up these great pillars north and south of the Earth's middle yet nearer to it than the chasm; and the Gods placed lamps upon them and the Earth had light for a while.
But the pillars were made with deceit, being wrought of ice; and they melted, and the lamps fell in ruin, and their light was spilled. But the melting of the ice made two small inland seas, north and south of the middle of the Earth, and there was a northern land and a middle land and a southern
The History of Middle-earth Volume 5: The Shaping of Middle-earth, The Ambarkanta, p. 238
(Interestingly, the appearance of inland seas after the destruction of the pillars was never changed, although the story of the ice pillars was abandoned.)
Angband and Utumno
If the pillars are too mountain-like to qualify, then it would probably be either Utumno or Angband, Melkor's fortresses, but there is no indication of their depth:
Now Melkor began the delving and building of a vast fortress, deep under Earth, beneath dark mountains where the beams of Illuin were cold and dim. That stronghold was named Utumno. [...]
The Silmarillion, Of the Beginning of Days, p. 29
Deep in forgotten places that cry was heard. Far beneath the rained halls of Angband, in vaults to which the Valar in the haste of their assault had not descended, Balrogs lurked still, awaiting ever the return of their Lord [...]
ibid., Of the Flight of the Noldor, p. 85
But at the last the gates of Utumno were broken and the halls unroofed, and Melkor took refuge in the uttermost pit. [...] Nonetheless the Valar did not discover all the mighty vaults and caverns hidden with deceit far under the fortresses of Angband and Utumno.
ibid., Of the Coming of the Elves, pp. 48-49
[...] and all the pits of Morgoth were broken and unroofed, and the might of the Valar descended into the deeps of the earth. There Morgoth stood at last at bay, and yet unvaliant. He fled into the deepest of his mines [...]
ibid., Of the Voyage of Eärendil and the War of Wrath, p. 303
On top of Angband was Thangorodrim, three peaks that were created by Melkor:
There he delved anew his vast vaults and dungeons, and above their gates he reared the threefold peaks of Thangorodrim, and a great reek of dark smoke was ever wreathed about them.
ibid., Of the Flight of the Noldor, p. 86
The Atlas of Middle-earth, pp. 14-15
Tolkien never specified the height of Thangorodrim, but he did make a drawing that showed both the island of Tol Sirion and the mountains in the distance:
Pictures by J.R.R. Tolkien, #36
Fonstad calculated that for this drawing to be possible, the mountains had to be about 35,000 feet (10 km) high. As a comparison, Mount Everest, the highest mountain on Earth, stands at about 30,000 feet (9 km). I somewhat question the validity of this estimate based on such an imprecise drawing.
In any case, if we take the whole thing as one massive structure, it would be the tallest ever built after the pillars.
That one is easy: Sauron's Barad-dûr. It was built in the Second Age after Sauron's return to Mordor. It is also probably the tallest freestanding structure on Middle-earth ever built.
Its height was never specified in any texts. When Frodo puts on the Ring to flee from Boromir, he ends up on Amon Hen and has visions of Barad-dûr:
But against Minas Tirith was set another fortress, greater and more strong. Thither, eastward, unwilling his eye was drawn. [...] Then at last his gaze was held: 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.
The Lord of the Rings: The Fellowship of the Ring, The Breaking of the Fellowship, p. 527
Whether Frodo is reliable at this point is unclear. If he is, then Barard-dûr would be higher than Minas Tirith, which stands at 1000 feet (300 m).
This is really unclear. I think it would probably be the Tower of Avallónë, where the master palantír was kept. It was the first thing sailors would see when they drew near to Aman.
[...] and there is in that land a haven that is named Avallónë, for it is of all cities the nearest to Valinor, and the tower of Avallónë is the first sight that the mariner beholds when at last he draws nigh to the Undying Lands over the leagues of the Sea.
The Silmarillion, Akallabêth, p. 310
[...] it is believed that thus [Elendil] would at whiles see far away even the Tower of Avallónë upon Eressëa, where the Masterstone abode, and yet abides.
ibid., Of the Rings of Power, p. 350
Other notable towers would be:
- Mindon Eldaliéva, the tower of Ingwë built in Tirion (where the Noldor lived in Valinor);
- Elostirion, one of the three White Towers built by Gil-galad for Elendil near the Grey Havens; and
- The Tower of Turgon in Gondolin. Fonstad says that "its turret [is] standing eight hundred feet above the Vale", but it seems to be based on a yet another very imprecise sketch, much like Thangorodrim above.
If we assume that the Orcs built the stairs to the Tower of Cirith Ungol, then that would be it, although they did not build the Tower itself. If not, then they probably contributed somehow to Barad-dûr, so they get to share the credit. Apart from that, I don't know of any other major structures built by them.
It was most certainly the Endless Stair that went from the bottom of Moria to the peak of the Silvertine (Zirakzigil), upon which was Durin's Tower. Dwarves also built other strongholds and deep mines before Moria, such as Nogrod, Belegost and Erebor, but it is unclear how they compare to it.
The Atlas of Middle-earth, p. 129
Fonstad says the guardroom was 3900 feet (1190 m) deep, but I don't know where she got this information. She has no references for this particular statement and I couldn't find anything else on it.
The travellers camped beside the archway in a guardroom with a well. The guardroom has been shown about 3900 feet deep, but there were hammer sounds heard through the well shaft from a lower level. The mines could have been as deep as 12,500 feet and still have been within limits reached in our Primary World.
The Atlas of Middle-earth, p. 128
Minas Tirith would probably be the tallest at over 1000 feet (300 m) high, but it was made by excavating the side of a mountain, so it is not a freestanding structure. The height from the ground to the highest level is 700 feet (210 m) and on the highest level is a 300 feet (90 m) tall tower, the Tower of Echtelion, also known as the White Tower.
Up it rose, even to the level of the topmost circle, and there was crowned by a battlement; so that those in the Citadel might, like mariners in a mountainous ship, look from its peak sheer down upon the Gate seven hundred feet below. [...] 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.
The Lord of the Rings: The Return of the King, Minas Tirith, p. 11
The Atlas of Middle-earth, p. 139
I think it's a tie between the Temple of Armenelos and Orthanc. Both are 500 feet (150 m), plus an additional component of unknown height. The Temple was built on Sauron's orders in Armenelos, the capital of Númenor:
But Sauron caused to be built upon the hill in the midst of the city of the Númenóreans, Armenelos the Golden, a mighty temple; and it was in the form of a circle at the base, and there the walls were fifty feet in thickness, and the width of the base was five hundred feet across the centre, and the walls rose from the ground five hundred feet, and they were crowned with a mighty dome.
The Silmarillion, Akallabêth, p. 327
So the walls are 500 feet (150 m) tall but there is also a dome on the walls, making the structure taller than this. If we assume the dome is a hemisphere, then it must be 250 feet (75 m) tall, but there is no information on its shape. As for Orthanc:
A peak and isle of rock it was, black and gleaming hard: four mighty piers of many-sided stone were welded into one, but near the summit they opened into gaping horns, their pinnacles sharp as the points of spears, keen-edged as knives. Between them was a narrow space, and there upon a floor of polished stone, written with strange signs, a man might stand five hundred feet above the plain.
The Lord of the Rings: The Two Towers, The Road to Isengard, p. 192
The height of Orthanc would then be 500 feet (150 m), plus the height of the "gaping horns". Whether the dome of the Temple or the horns are taller is unclear.