Just like the tallest building changes from one building to another so does the tallest mountain changes from one mountain to another but while one is man made, the other is of natural occurrence. You might think that mount Everest is the only tallest mountain but this might change tomorrow. Remember that the earth isn't flat, and it isn't completely round while still moving.
In 1792, Mount Chimborazo was tallest mountain in the world, in 1808, Mount Dhaulagiri was the tallest mountain and the title also changed hands to Mount Kangchenjunga in 1847 and after that, mount Everest became the verified tallest mountain just about 5 years after Kangchenjunga in 1852.
Although Everest has held the title of the tallest mountain, its measurement has changed over time and this is because people have different definition for categorizing mountains. Also, technology has a hand in the measurement difference just like the mountain also ha a role to play because they are changing even though they look like they are unchanging.
You must have learned in school that the earth crust is divided into plates and for a mountain to arise, it occurs when tectonic plates bump, grind, and spread apart also known as Convergent, Transform, and divergent respectively. These phenomenon can change the height of a mountain depending on when it was measured. Beneath a mountain are huge currents of lava that carries molten rock to the surface to increase the height and width of mountain. Plates also move when denser crust sinks under lighter crusts until there is pressure leading to a volcanic eruption and a volcanic mountain.
Talking about plates, Mountains like Everest and the rest of the Himalayas started growing into a mountains when the Indian Plate and the Eurasian plate slammed against one another about 50 million years ago and it still continues. The processes that make mountains can also break them as rocks can break when plates collides and eroded by weather or nature. Since the mountain has lava, then mountains can start to sink gradually if they are heavy enough preventing them becoming higher. An example of this sink happened in 2015 when a magnitude of 7.8 earthquake happened in Nepal making so many peaks in the Himalayas shorter by about 60 centimeters.
While Everest is growing yearly and holds the title of the tallest mountain, it might not be long before it loses that title because mountains like Nanga Parbat in Pakistan is growing faster yearly than Mount Everest although it isn't as tall as Everest yet but if it continues at this rate, it might overtake Everest in a few 100 thousand years from now.
Talking about tallest, if we measure height from sea level, then Everest is the tallest but if we were to measure mountain from ground to top from land then mount Denali would have this price but if we were measure a mountain from base to summit han Mount Mauna Kea would have held a title but more than half of it is submerged in water.
Sea level is another debate because sea level differs depending on the area of the earth. The radius is water level bulges around the equator compared to other parts of earth which means that the radius of sea level in Ecuador and Antarctica are not the same but for us to have an equal sea level, the mean sea level is said to be at 0 level elevation. If you want to be closer to stars, standing on mount Chimborazo will get you closer.
Deciding on the tallest mountain based on mean sea level would mean that Everest is the tallest mountain but if we were to pick from the base at different criteria, then there will be so many mountains dragging for the throne but then were do we define as the base to start measuring?
Reference
https://www.nature.com/articles/ngeo2623
https://www.egu.eu/newsletter/geoq/09/articles4.pdf
https://www.sciencedirect.com/science/article/abs/pii/S1342937X18302041?via%3Dihub
https://www.dos.gov.np/everest/downloads/Nitin_Joshi.pdf
https://ia601908.us.archive.org/24/items/Everest_201304/everest.pdf
https://oceanservice.noaa.gov/facts/highestpoint.html
https://www.ijcoe.org/article_155187_136c0f16036a35039cc2449b7fad1707.pdf
https://www.bbc.com/future/article/20220407-how-tall-will-mount-everest-get-before-it-stops-growing
https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol50_2015_4_Art07.pdf
https://pubs.usgs.gov/gip/dynamic/himalaya.html
https://www.ultimatekilimanjaro.com/mauna-kea-the-tallest-mountain-in-the-world-underwater/
https://www.egu.eu/newsletter/geoq/09/articles4.pdf
https://www.sciencedirect.com/science/article/abs/pii/S1342937X18302041?via%3Dihub
https://www.dos.gov.np/everest/downloads/Nitin_Joshi.pdf
https://ia601908.us.archive.org/24/items/Everest_201304/everest.pdf
https://oceanservice.noaa.gov/facts/highestpoint.html
https://www.ijcoe.org/article_155187_136c0f16036a35039cc2449b7fad1707.pdf
https://www.bbc.com/future/article/20220407-how-tall-will-mount-everest-get-before-it-stops-growing
https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol50_2015_4_Art07.pdf
https://pubs.usgs.gov/gip/dynamic/himalaya.html
https://www.ultimatekilimanjaro.com/mauna-kea-the-tallest-mountain-in-the-world-underwater/