Suppose there was a huge crater on Earth (and we were alive to be there). That the rim was as high above the floor as gravity will allow- let’s say 7 miles- and the crater was broad enough that the floor followed the curvature of the Earth. At a certain width a line of sight between opposite sides of the rim would just clear the floor. How wide would that be?
If you diagram it out, you would get an isosceles triangle with sides r+7 and altitude r where r is the radius of the Earth in miles. The distance from one rim to the other in a straight line is 2 sqrt(14r + 49) or approximately 471 miles.
It’s the same question as 2 mountains, each 7 miles high, with sea (or sea-level land) between them. No crater needed!
Using Pythagoras and a radius of 3961 miles I get the same answer of 471.2 miles between the 2 peaks.
Now, I think the actual distance is further because of atmospheric diffraction (which, if I remember correctly, will make the distant mountain appear higher).
And to generalize, any body of radius r and a crater (or mountains) of height h, the line of sight where the other rim (or peak) is at the horizon is 2 sqrt (2hr + h^2) so two Olympus Mons on Mars just on the horizon from each other are about 773Km or just under 480 miles apart.
You’re right; I don’t remember now why I thought a crater was necessary.
The ultimate sniper’s challenge. 
A crater could help by eliminating some of the curvature of the earth that got in the way of your sight lines. For example if it was 7 miles deep, the problem would change so the you effectively had a earth with radius r-7 and 14 mile tall mountains. Using Lumpy’s formula this would increase your sight range to 666.3 miles.
Nitpick: Atmospheric refraction
Hopefully I didn’t misclick while working this out in Excel.
Per Wiki the formula for the ~distance in miles to the horzon is:
1.22 x sqrt(height in feet of point of view)
Ergo at the top of a 7-mile high mountain the distance to the horizon is 1.22 x sqrt(36960) = 1.22 x 192.2498 = 234.5 miles
I take that to mean the summits of two 7-mile high mountains would become visible to each other at a distance between them
of 234.5 miles x 2, or 469 miles.
Presumably, by the “on earth” in your OP you mean “between any two points on earth”.
Because otherwise, there are lots of answers.
standing here on earth, I can see the moon, thousands of miles away. Or the sun, millions of miles. Or various stars & astronomical bodies even further away.
(The Andromeda Galaxy (M31), at 2.25 million light years away, is usually the furthest object visible to the naked eye (but that takes better eyesight than mine). There have been others farther away (like a thousand times farther), but those have been supernova explosions, and are usually only visible for a few hours or days.)
This website is worth wasting a few minutes on. Apparently, Columbia is home to this particular geographical oddity.
Why would it have to be a “floor” per se..what happens with a conical void shape between the mountains–how deep could it get before the mass of the earth started pushing it back together? If a cone shape could go all the way to the center of the earth, the answer is “darn near the diameter of the earth” but of course gravity will not allow this.
Months, actually. Supernovae stay near their peak brightness for several months.
Great site, thanks! Nitpick: Colombia, not Columbia.
Your welcome.
Homonyms (I think that’s the term) are the bane of my spelling. 