… Why did the Imperialist Britain / Colonial American armies/governments stop where they did? It looks like in 1970 we traded off negative seven-hundred acres to Mexico (but what about the people!?); since then, (as well as for thirty years prior), there have been no significant territorial changes. Why did we stop? Why does any empire stop?
The British Empire stopped because after taking French lands in North America, it butted up against the Spanish Empire in the south, while unexplored lands further north were very cold and not of much use. In the 19th century the United States did forcibly annex most of Mexico. We probably would have taken the rest of the country if the federal government saw any advantage in doing so. And while the United States hasn’t formed any new states in Latin America during the 20th century, we certainly haven’t hesitated to send troops (or CIA-supported thugs) into many countries down there.
There is an interesting footnote worth mentioning about Cuba. During the 1850’s, the United States had the opportunity to purchase Cuba from Spain for twenty million dollars. However, Congress was afraid that the very corrupt President Buchanan and his cronies would find some way to steal the money, so the purchase was never approved.
well, the imperial armies of the United States twice tried to add Canada to the empire, and were twice rebuffed.
I think you mean, “Tried to liberate Canada from the oppressive British Empire” there. 
I’m pretty sure the whole Cuba matter happened during the Pierce administration…Buchanan was, at the time, the US Minister to Britain, and one of the people involved with the treaty. One of the big problems was the slavery issue, and Northern senators didn’t want to add more slave territory to the country.
We seized land in areas with low population densities, inhabited by peoples with no nation-state-type organization or advanced technology (rifles, railroads, etc.). To the north, we ran up against an area of low population density, but one that was already taken by another high-tech, state-organized, First World society (and, during our most expansionist period, was part of the most powerful empire on the planet). As has already been pointed out, we tried to take Canada, but the British Empire wouldn’t let us.
To the south, we’ve got large, dense, urbanized populations–when we took half of Mexico, we took the relatively unsettled half–and while they may not traditionally be regarded as being our military equals, Latin Americans states are nonetheless states, with standing armies, guns, and so on.
We also expanded overseas, but, apart from managing to absorb the Hawaiian Islands, and semi-absorb a few minor territories, there weren’t many areas nearby that were conveniently “uninhabited” (i.e., inhabited by peoples with radically lower levels of military technology and political organization). When we took over the densely populated Phillippines, we weren’t able to simply wipe them out or send them off to Oklahoma and (for cultural and frankly racist reasons) we didn’t want to simply assimilate them and convert them into American citizens (nor did the Filipinos wish to be so assimilated).
Basically, we kept taking over new areas until we ran up against people too strong to dispossess in every direction, and ran out of new frontiers to conquer.
Were they rebuffed, or once they got there, looked around, didn’t find anything to their liking, and decided to head back home? 
Why doesn’t the “United States of America” encompass all of North and South America?
Just be patient and give it some time.
The OP might find the wiki article on “Manifest Destiny” to be of interest.
Sure, that’s what armies always say when they come home from a butt-kicking. 
I can’t give a cite, but I know back in maybe the 1970s or 1980s, I read a survey that supposedly showed most Candians thought their country would be part of the US in a generation or so.
Unlike our good wholesome American grapes, the grapes in Canada are very sour, so nobody would want them anyway.
[nitpick]To describe the British Emprie as “First World” is an anachronism. People didn’t start allocating countries to one of three “Worlds” until the Cold War.[/nitpick]
I always thought the giving away the Panama Canal Zone was a big mistake. It always seemed to me like a good place to draw a line on a map.
And even then, we invited them to join us. The first US governing document, the Articles of Confederation had a provision allowing Canada to become a US State whenever they chose to. Any other territory had to ask Congress to allow it to become a State.
To clarify, that was “Canada” meaning southern Ontario and southern Quebec, the British colony > province of that name, not the ten-province contitinent-spanning nation of today. Roughly, the area between North Bay and Baie-Comeau, plus the Gaspé, south to the US border and excluding New Brunswick.
To add to what Polycarp is saying:
Quebec under the Royal Proclamation of 1763;
Quebec under the Quebec Act of 1774, enlarged by much of what is today Ontario, Labrador and the American Midwest. Of course, the Americans didn’t recognize Quebec’s sovereignty over the Midwest as some American states also had claims over it, and had Canada decided to join the American Revolution, it would presumably have had to cede it.
Probably. Most of the states ceded their territory in the midwest to the national government. the state of Canada might have been able to hold onto some of it, though.
I’m just going by the writings of P.J. O’Rourke, but according to him, we took the more industrialized half of Mexico. He asks a Mexican politician why Mexico still has a grudge against the USA, and he replies, “You took half our country. And what’s more, you took the half with the paved roads.”
Umm no. The area wasn’t even very Mexican at the time. There were a mere 80K Mexicans in the areas ceded under the Treaty of Guadalupe Hidalgo, and they constituted a mere 20% of the population. That area had only been part of Mexico for about 20 years.
And there were no paved roads, the first macadam road constructed in America ( in State of Maryland) was in 1823. The El Camino Real was dirt during this period.