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Throng La Pass on Annapurna trek in Nepal, 5416m.
Went over 5k meters a few times on the Ausangate trek in Peru as well. 6k is on my bucket list, but I'm not sure where would be the best option that is not too technical. I know that there is volcano hike near Arequipa, Peru, which takes you that high, but I'd rather see a different country.