A question I find myself asking over and over as I'm playing Civ V is how many cities I want to found. Way back in the 'good old days' of earlier Civ games the answer was as many as I could get my grubby little hands on. More cities were just better! Each new city that you founded would give your empire more money, more research, and another place to build military units. At some point happiness and corruption would become an issue, but because both happiness and corruption were local to each city the worst that would really happen is the newest cities would suck for a while. The rest of your Civ would be chugging along just fine. More recently they've added mechanics to punish you for making too many cities. Now it's no longer just awesome to have more cities. It may even be a big problem to get extra cities. The question I have is where is the line between awesome and terrible? To answer that I need to know what changes when you found an extra city.
First off, social policies become more expensive the more cities you have. The amount it goes up is based on the size of the map and is somewhere between a 30% or a 15% boost in cost per extra city. This cost increase is additive, not multiplicative. Unfortunately a lot of the culture gain in the game doesn't scale with number of cities at all. You can only build one copy of any given wonder, after all. City states only give you a flat amount of culture, not an amount conditional on how many cities you have. In my current game I'm getting 18 culture from happiness, 214 from city states, 99 from a golden age, and only 263 from cities. A given new city would be able to bring in 16 culture total if I built every possible culture building I can currently build, including the one that would require the new city to convert to a specific religion. That's about a 2.7% increase in overall culture generation. I currently have 4 cities, so it would increase the cost of a new one from 1.45x base to 1.6x base, or a 10.3% increase in cost. So founding a new city would be a big set back on getting a new social policy. Puppet cities, on the other hand, don't increase the cost but do provide culture so they're a strict upgrade. Not a very good one (the puppet is unlikely to build all those culture buildings right away) but still a net positive on social policy cost.
It also turns out research gets more expensive as you get more cities. It's a 2% increase per city, and looks to include the first one. Puppeted cities are counted against you this time. Just like with culture there are non-scaling sources of research. 241 of my 1122 research is coming from city states, for example. An awful lot of it is coming from one city I build to be the super science city. 5 great scientists have built buildings in the area, and it has an observatory and the multiplier national wonder. I am getting 2 science per specialist though, so any new city that grew to a reasonable size would be adding a fair amount of science. My worst city (size 23) is making 114 science per turn and has nothing special going on. Another copy of that city would be a 10% boost in science earned while only a 1.85% boost in science cost. It would slow me down a little while the new city got up to speed and built all the buildings and grew big enough to have a lot of specialists to be fair, but once it got running it would be a net positive unlike with social policies where it would just be terrible. Puppets here are probably going to be worse since they do increase cost and I can't direct them to build the right buildings to either grow fast or make lots of science. And if I didn't have the rationalism social policy tree it would probably end up hurting since the AI likes to use specialists and they're bad for science without those policies.
Happiness can be a big problem with number of cities. You get 3 unhappiness for every individual city, and 1 unhappiness for every citizen in a city. There are lots of buildings that can help offset that happiness problem, and if you build beside a luxury good you get 4 free happiness right off the hop. Religion added in some new ways to get happiness. I feel like I need to build a bunch of buildings and pick up some social policies, but that adding an extra city beside a luxury good would be a net positive for happiness. It's something that needs to be managed, but it can be done. Even without a new luxury having the people spread out is better than all together because they can build two colosseums if they're in two places!
As far as I can tell everything else gets better the more cities you have. You just earn more faith by having more cities. You can produce more things at the same time. You earn way more money by having more cities, especially if you focus them on making money. You can build more military units at once. You can grow your population faster in multiple spots than in one.
Most of the time I'm playing I care about social policies. They just seem cool, and I want as many of them as I can get. So having fewer cities feels better to me. I need enough cities that I can build ALL the wonders to keep the culture production flowing, but tacking on extra cities just seems bad. Beyond social policies it seems like lots of cities could work just fine. Maybe I should try a game where I build a bunch of them to see what happens. I fear that I won't be able to get the social policies that give the happiness in time and I'll just end up a very unhappy civ though.