In Middle-earth throughout all time it is almost certainly Thangorodrim:
And looking out from the slopes of Ered Wethrin with his last sight he beheld far off the peaks of Thangorodrim, mightiest of the towers of Middle-earth...
The Silmarillion, Chapter 13: The Return of the Noldor
In this case "towers" almost certainly means mountains. Morgoth almost certainly raised them in opposition to the great Taniquetil, home of Manwë.
There is no indication of what was the tallest mountain in the Third Age but of those listed below, I would hazard to guess that the Mountains of Moria, and therefore Caradhras, were the tallest based purely on the fact that the Great Eagles had their Eyries in those mountains and I could only imagine the servants of Manwë choosing the loftiest peaks (although they were located on Celebdil not Caradhras).
The aforementioned Taniquetil is the tallest mountain in Arda and is so described in the Silmarillion:
Manwë and Varda are seldom parted, and they remain in Valinor. Their halls are above the everlasting snow, upon Oiolossë, the uttermost tower of Taniquetil, tallest of all the mountains upon Earth.
Other notable mentions in Middle-earth (and Númenor) include the Crissaegrim peaks, the Meneltarma, Orodruin, Erebor and Caradhras
The Crissaegrim peaks were the location of the Eyries of Thorondor the Greatest Eagle:
[Húrin] descried far off amid the clouds the peaks of the Crissaegrim ... Then Húrin looked up to the grey sky, thinking that he might once more descry the eagles, as he had done long ago in his youth; but he saw only ... clouds swirling about the inaccessible peaks, and he heard only the wind hissing over the stones.
The Silmarillion, Chapter 22: Of the Ruin of Doriath
The Meneltarma, the Pillar of Heaven, reigned supreme on the star-shaped isle of Númenor. While a specific height is not given the tower is often described as being of a great height, tall enough that the exiled Faithful would sail out in search of it's tip that was storied to still poke out of the sea as an island.
in the midst of the land was a mountain tall and steep, and it was named the Meneltarma, the Pillar of Heaven, and upon it was a high place that was hallowed to Eru Ilúvatar
Among the Exiles many believed that the summit of the Meneltarma ... was not drowned for ever, but rose again above the waves, a lonely island lost in the great waters; for it had been a hallowed place, and even in the days of Sauron none had defiled it.
Orodruin, Mount Doom:
Still far away, forty miles at least, they saw Mount Doom, its feet founded in ashen ruin, its huge cone rising to a great height, where its reeking head was swathed in cloud.
Return of the King, Book VI, Chapter 2: The Land of Shadow
... and now they could all see the Lonely Mountain towering grim and tall before them.
The Hobbit, Chapter XI: On the Doorstep
And last but not least, Caradhras. While there's speculation that Celebdil might have been taller as that is where the Eagles' Eyries could be found, Caradhras seems to be the clear winner according to the Lord of the Rings:
At the left of this high range rose three peaks; the tallest and nearest stood up like a tooth tipped with snow; its great, bare, northern precipice was still largely in the shadow, but where the sunlight slanted upon it, it glowed red.
Fellowship of the Ring, Book II, Chapter 3: The Ring Goes South
And later in the same chapter:
On the third morning Caradhras rose before them, a mighty peak, tipped with snow like silver, but with sheer naked sides, dull red as if stained with blood.
While no heights are ever really given for the mountains, and idea of the height of Caradhras can be gathered from this answer on the height of the structure of the stairs and this answer about how big Khazad Dûm was.