Cities can use a 36 hex grid at maximum.
It seems to me that 3 is also the maximum "range" at which one can purchase new tiles. Is this correct, or is it possible to buy land further off?
No, you cannot buy tiles outside the city radius of 3 tiles. You can use a great Artist to culture bomb those tiles if you need the strategic/luxury resources there, but don't want to build a city.
The city will continue to expand through culture up to a maximum radius of 5 tiles. But you need a lot of culture for that, the Angkor Wat wonder in that city is very useful for that.
You can never work tiles outside the 3 tiles radius, but you can get the resources on those tiles.
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@Oak you only need to improve a strategic or luxury resource to add it to your trade network. you do NOT need to have a citizen work that tile to gain access to that resource. If you are unsure, just check your happiness icon on the top left to see which resources (luxury) you have access to, or look at the strategic resources listed to the right of that.
sorry that should have been a comment not a reply