This is a work in progress
I've looked for the tallest structure made by every race, although it's difficult to come up with even an approximate height for some of them.
Depending on your definition of "structure" and "mountain", the tallest thing the Valar built was probably the pillars:
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.
It is not entirely clear what the pillars are made of. Karen Wynn Fonstad, in her Atlas of Middle-earth, 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 land.
Interestingly, the appearance of inland seas after the destruction of the pillars was never changed, although the story of the ice pillars was abandoned.
If the pillars are too mountain-like to qualify, then I'm not sure. The Valar were not great builders of structures, although they did create a lot of mountains. The highest structure would be Ilmarin, the mansions of Manwë and Varda, perched on top of Oiolossë, the highest peak of Taniquetil.
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 structure on Middle-earth.
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.
[...] 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.
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.
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.
I think it's a tie between the Temple of Armenelos and Orthanc. Both are 500 feet, 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.
So the walls are 500 feet tall but that there is also a dome on the walls, making the structure taller than this.
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 height of Orthanc would then be 500 feet, plus the height of the "gaping horns". Whether the dome of the Temple or the horns are taller is unclear.