can a city buy a city?

It is possible, but most times a city will "annex" surrounding area, which means expanding the city limits to include that area. I don't know the specific legalities involved in annexation. Regarding buying a city, I know it happened in Florida, where a very small town (the town was actually mostly a small amusement park) was bought by the larger city next to it.

In order to buy a city, there would have to be a city that's for sale. I don't believe that kind of thing happens. The only way someone would be able to sell a whole city is if they owned every piece of land in it. That's not how things work. Individuals and corporations own most of the land and buildings in a city. The city itself owns a very small percentage of the total area.

Generally, cities grow by annexation and merger, not purchase. It sounds like you are asking how two smaller cities could combine to form one larger city. For a recent example of where this happened, look up Helena-West Helena, Arkansas on Wikipedia. These two small cities decided to merge because they felt it would lead to a more efficient government.

Sure that's why Chicago has so many distinct neighborhoods, many years ago, they often annexed other towns. I live in one of them; Lakeview used to be a separate town; now it's a neighborhood of Chicago.

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