I estimate 400 km.
The Earth’s radius is 6370 km. If we assume Everest at 8.88 km altitude is more or less surrounded by a plane at 3 km altitude (i.e. Everest is 5.88 km above the plane), its horizon is 274 km. If another mountain at 8 km altitude can just be seen over that horizon it is 252 km away. This gives a possible distance of 252 +274 = 526 km. Since it is unlikely to have 2 of the Himalayas (or other mountain ranges) set up perfectly for that, I will say the furthest distance on Earth is 400 km

I estimate 400 km.

The Earth’s radius is 6370 km. If we assume Everest at 8.88 km altitude is more or less surrounded by a plane at 3 km altitude (i.e. Everest is 5.88 km above the plane), its horizon is 274 km. If another mountain at 8 km altitude can just be seen over that horizon it is 252 km away. This gives a possible distance of 252 +274 = 526 km. Since it is unlikely to have 2 of the Himalayas (or other mountain ranges) set up perfectly for that, I will say the furthest distance on Earth is 400 km

Some excellent thinking there.

(Though we suspect you meant plain instead of plane.)

But as it turns out, the longest line of sight DOESN’T involve Everest.

It’s from Hindu Tagh in China to Pik Dankova in Kyrgyzstan, at 538km.