Civilization – In Civ5, is there any point to immediately annexing a city

A few options:

1. The most obvious solution is to increase the production of the city. A few ways to do it are:
   • Build a few production-increasing buildings before you build your long building.
   • Purchase production-increasing buildings with gold.
   • Increase the production yield of the city's tiles by improving them with workers, or build factories using great engineers (though great engineers can be used in a different way, see below). Marble is especially useful, as any city which has a worker on improved marble gets a 25% production boost, though only for wonders.
   • Shoot for technologies or social policies that increase production. For example Steam Power increases production from lumber mills, so if you have a lot of those around your city this technology could really give it a boost; Railroad will allow you to connect cities by railroad which increases production; and the Order social policy will increase all production for building by 25%.
   • Redirect the civilian allocation to increase production. You can just use the radio button under the "civilian allocation focus" to let the computer choose the best allocation, or you can do it yourself manually. It means that you should put civilians inside tiles / buildings that have the most impact on production. If you're desperate for production you can de-allocate a citizen altogether: each unemployed citizen produces 1 production per turn.
2. Chop down trees around the city. Chopping down trees gives a small boost towards the currently in-progress building. You can even chop trees in an area outside your territory, it still gives some production (though less, and it gets further decreased with distance) to your closest city.
3. The great engineer can speed up buildings and wonders. When you use his ability it gives a tremendous boost to the current in-progress structure, often making it complete just one turn after you use his ability. Very powerful, don't waste it on anything which isn't worth it. Notice that if you build something ridiculously difficult it will probably not get down to 1 turn.

Note: if it's particularly the courthouse which is giving you trouble, consider getting the Police State social policy, under the Autocracy tree, which makes occupied cities produce only half the unhappiness value.

The diplomacy summary that you get from clicking the scroll in the upper right will show you all the civs and city states at once, as well as what resources they have available, how much gold they have, and how much gold per turn they are making.