What is the tallest fully underwater seamount in the world?
According to the Oxford Dictionary of Earth Sciences,


  seamount Isolated, submarine mountain rising more than 1000m above the ocean floor. The sharp, crested summits of seamounts are usually 1000-2000m below the ocean surface. Seamounts are of volcanic origin.


To avoid classifying seamounts by arbitrary sea level (dependent on availability of surface water), the key point is then that seamounts are features of volcanic origin that rise over 1000m above oceanic crust.

The Hawai'i-Emperor seamount chain is of volcanic origin; all of the islands in this chain are seamounts. Mauna Kea only rises 4207m above sea level - but measured from its base on the oceanic plate it is 10100m high, much taller than Mt Everest. Mauna Kea is - pretty conclusively - the highest seamount in the world. The Woods Hole Oceanographic Institute agrees.