The Navel Gym isn't a building, and instead spans the entire mountain on Navel Island. Danny makes Trainers participate in several mountaineering challenges, including scaling the mountain without the aid of any Pokémon. Natural geysers must be frozen and crafted into bobsleds for racing down the mountain. There is no battling involved at this Gym.
Pokémon used in Gym
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Danny's Nidoqueen was used by him in the first challenge against Ash. Using Ice Beam, she was able to quickly freeze the water coming out of the geyser in order to beat Lapras. Nidoqueen's physical strength was then used to help carve out a sled from Danny's slab of ice in the second challenge.
Scyther was the only one of Danny's Pokémon to be in both the second and third challenges. It first helped carve the block ice into a sled after which it then helped steer that sled in the race down the mountain.