I just saw this picture in Learn Something Every Day and thought of you. Some of the sites I've found reason it with gravity but doesn't explain very well.
How exactly the different intrinsic and extrinsic factors interplay to shape real mountains is an active field of research. Thus, it's not possible to say exactly how high a mountain could become on earth. However, there are several limits to that.
First, there is the issue of rock stability itself. Rock has a limited compressive strength, but quite a bit of weight (relative rock density is on the order of 2.5), so if a mountain becomes too high, the rock at the base will simply crumble or melt from the pressure.
Terzagi (1962); Géotechnique, Volume 12, Issue 4, pages 251 –270 calculated the theoretical height of the tallest vertical cliff as
In reality, these numbers are not readily achievable on earth. There are numerous intrinsic factors that limit rock stability - cracks, folds, etc., as detailed in e.g. Cruden (2003). The shapes of cold, high mountains in sedimentary rocks. Geomorphology 55:249, or in Schmidt and Montgomery (1997). Limits to relief. Science, 270:617.
Furthermore, it has been argued in Brozovitch et al. (1997). Climatic Limits on Landscape Development in the Northwestern Himalaya. Science 276:571 that it is really erosion through glaciation that ends up limiting mountain height.
This has recently been supported by Egholm et al. (2009). Glacial effects limiting mountain height. Nature 460:884, who do a more large-scale analysis. The abstract of this paper which summarizes the above much better than I do:
Here's the link to ref#5, which doesn't unfortunately, calculate the maximum theoretical height of a mountain. I guess geologists may mention these things in talks, but not in high-end journal publications.
In summary: The 15km limit may be plausible, but it's unlikely to ever be attained by real-earth mountains, even the 10km ones that hide from most of erosion in the sea.
An engineer and instructor from NASA posting on Quora had this to say about maximal height of mountains on Earth:
This answer would seem to confirm Jonas's explanation. However, it does not address some exogeological formations, such as Olympus Mons on Mars, which is approximately three times larger than similar formations found on Earth even though Martian gravity is about 53% that of Earth's. This seems to indicate that the relationship between gravity, geology, and maximal height may be too complex to be accurately summarized in a single rule of thumb or simple equation.
The original post on Quora can be found here, but a Quora account is necessary in order to view it.