Can "mountain climbing" describe the activity of walking up a mountain without any special equipment?
And how about mountaineering?
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According to britannica.com and the general google definition, mountaineering and mountain climbing mean :
The sport or activity of climbing mountains
So they can describe climbing up a mountain without equipment.
Personally, I think both of those words imply that there's special equipment being used. Mountaineering is also more formal than mountain climbing.
If you want to say someone climbed a normal mountain without equipment, you could use 'hiking'. If this is an extreme case, like someone climbing Everest without equipment, you'd want to specify that there's no climbing gear involved.