Question: Instead of discovering that earth was round, we find that we live on a infinite, flat "earth".

What are some cultural implications if we were to live in a world like that (disregarding any physical impossibilities, and that gravity, biological/ecological systems remain roughly the same as reality).

One interesting idea is that cities could be tens of thousands of kilometers from another and that one can only explore so much of the world in their lifetime.

EDIT: I don't mean flat as in geometrically flat, but flat in general. There would still be mountain ranges and oceans and forests and stuff. But there would be different and exotic ecological systems as you travel.

Answer: 
The world would become progressively more alien as you moved across it.

Unlike on Earth, where animals and humans can move across the entirety of the world given a few thousand/million/tens of millions of years, an infinite flat earth would always have somewhere that's far enough away that nothing from where you are has ever been there.

This would probably lead to a continuous cycle of mass extinctions brought about by animals migrating in from a place where they have evolved in a manner that allows them to out-compete their neighbors. Something like Europe could exist, with mammals and birds forming most of the terrestrial fauna, when a group of highly evolved velociraptors finishes ten million years of gradual migration and then decimates the entire local ecosystem.

At a great enough distance, *nothing* would have even a single common ancestor. You'd likely get great patches of land in which only single celled or incredibly basic forms of life exist, since this form of life evolves in abiotic conditions relatively quickly, but then takes billions of years to become multi-cellular. These stretches would separate islands of complex life, which would become increasingly complex and highly evolved at their centers.

Aside from the mass extinctions that would occur when these life-islands grew into each other, different life islands could also evolve their own sentient species. These species, again, wouldn't share any common ancestors, and could well be separated by hundreds of millions of miles. They would all be incredibly alien to one another, more different than humans are from plants or mushrooms, but they could eventually come into contact with one another. Physically reaching one another would take quite a bit of time, since they'd likely be separated by immense distances, which would be difficult to traverse, but across which radio waves could probably travel.
Not good (or at least, not much better than our history).

The issue is not only as much "real estate" you have, but how much of it is usable. For example, most of the Earth surface has no human population (the oceans, Antartica) or just a bare minimum (deserts, Amazonian).

In that sense, if your area becomes overcrowded, maybe the nearest free soil is too far away (or has to pass through too many of other tribes'territory) to be of any actual interest.

So you would begin with isolated human groups, which would grown and expand. At the beginning, most of them would be able to send excess population out of the frontier, but with time the "frontier" tribes will find that the free soil is too far away (think how once, the USA frontier run through the Appalachian Mountains).

As technology progresses, the "frontier tribes" would become bigger and bigger (because with better technology is easier to keep the political unity). There would be cycles of frontier nations becoming too big, desintegrating in several nations, and the new "inner" nations using their power to push back the weaker nations of the privous cycle. Probably, sometimes population pressure in the older (inner) nations would lead to massive migrations/waves of conquests, like those of the Germanic peoples or the Mongols.