Taniquetil is the dwelling place of the Valar and the highest point on all Arda and tallest mountain on Middle-Earth. Is there any insight on what the height of Taniquetil is?
The precise height is never revealed; we just know that it's the tallest...anything in all of creation.
In Atlas of Middle-earth, Karen Wynn Fontsad speculates that Thangorodrim, serious runner-up for title of "highest mountain in Arda", was 35,000 feet high:
Prior to The History [of Middle-earth] the only references [to Thangorodrim] were the text and a drawing1 by Tolkien that showed the central peak in the distance. The text stated that these were the "mightiest towers of Middle-earth." [...] In the drawing, the central tower, as seen from the Pass of Sirion, appeared immense - far higher than the Ered Engrin. It would have to have been some five miles in diameter at the base and some 35,000 feet high!
The Atlas of Middle-earth Part 1: "The First Age" "Thangorodrim and Angband"
Assuming she's correct about this2, Taniquetil must be higher than 35,000 feet. But that's about the best we can do.
1 This drawing is a 1928 pencil sketch titled "Tol Sirion". A coloured version (not coloured by Tolkien), included below, was printed in The Silmarillion Calendar 1978
2 And bear in mind that this is Fonstad's intuition, based on her expertise as a cartographer and interpretation of a (very old) pencil drawing. It may or may not be anything close to correct