* If you consider any elevated landform a mountain: The shortest mountain would likely be a small hill, perhaps only a few meters high.
* If you use a more specific definition of a mountain: Many people consider a mountain to be a landform that rises at least 1000 feet (304.8 meters) above its surroundings. In this case, the shortest mountain would be something like Wycheproof, Australia, which has a peak elevation of only 43 meters (141 feet) above sea level.
However, there's no official, universally accepted definition of a mountain. So, the answer depends on your own definition!