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: 
Any world where an immortal can travel forever without repeating themselves has some interesting implications, some of the things to think about include the migration of disease strains through various populations, plant, human and animal, that call this world home. What happens when nomadic tribes that haven't stop moving in generations run into areas of closely settled farming communities where no-one goes anywhere and haven't for generations either. The migration of new plants and animals across the territories of the universe, introduced pests, the existence and extent of natural barriers like oceans and mountains is important as well. In fact in this kind of universe the water/land balance becomes of vital importance to how things move, if all bodies of water are ultimately landlocked then they don't provide a barrier to migration only a detour, if on the other hand all continents are ultimately islands then migrations are heavily curtailed in the overall scheme. Oceans may be too large to navigate, rivers too vast to bridge, deserts too wide to cross, any geographic feature that can act as a barrier or even a resource can potentially be so large as to create problems for mortals. A wood cutting camp may disappear into the forest they're cutting as regrowth cuts them off from their compatriots who still get logs floated down stream to them but have forgotten from whom they come. Human memory itself becomes a problem in dealing with such large distances.

How far and how fast waves of civilisation grow and collapse across an infinite space is also important; start with the speed of communication and transmission of force, whether aimed at revolting peasants or securing a border, this really sets some limits on centrally organised cultures. A traveler from a civilised enclave will probably be out of touch with their birth culture rather rapidly and into truly alien cultures reasonably shortly thereafter. In theory you could have a culture that spreads at the same rate it fails creating a ring of cities and industry that expands across the world leaving a growing circle of ruins in it's centre. Also consider what limits there may be on technological growth imposed by resource availability, if the world is flat is it also thin and therefore relatively devoid of mining opportunities 

The other thing to look at is kind of the opposite of all of the above it's about things that are similar not things that are different. Ultimately in a truly infinite space you will leave behind your cultural, and possibly even your genetic, heritage but it's important when working on a world so vast to think about how far certain practices an individual takes for granted *are* distributed, can an immortal find that though they've walked 50 miles a day for millennia they still find that everyone they meet grows wheat and bakes bread or can a man walk but three days up the road and find the world is unrecognisable? How far can you go and *not* notice the differences in how the world is "out here".

That's just a few thoughts from a piece I've been working on, on and off, for a while, that uses Cosmic Macaroni as the backdrop but the principles are the same. I'm going to have to think about uses for an infinite flat space I've done "humanly infinite" spaces and objects before (worlds so large that nothing living in them lives long enough to see any noticeable fraction of them) but never flat spaces.
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.