I know that Granary is compulsory in every city, but what if you're playing a non-Creative leader and wanted to grow a city's BFC? Do you build Theatre? Library? Monument?
A few options. My favourite early game is to wait until 1 turn before the city grows from 1 to 2 and start building a monument, then whip it next turn for 1 pop.
Chopping one forest for a monument can get it out pretty quick, especially if you work a plains forest or something else that makes decent hammers until it's done. If you have a good food source it's better to do the former though, and use the forest for something more useful.
I don't usually build stonehenge, because I don't usually want GPs (I let the AI found religions and then take their shrines). The times I do are when I'm far from boxed in and I know I'll be building at least 5 or 6 cities peacefully, and I have lots of forests and/or stone and/or an industrious leader. In my opinion the hammers are usually better spent on more settlers/workers.
For my interior cities I use libraries, since I almost always want the library anyway, and getting culture out instantly isn't as important for seizing tiles from the enemy (or blocking areas off). I like the library over theatres.
With a state religion and a missionary missionary, if I'm settling away from rivers, roads, or coasts, I drop the missionary in right away to get the +1 culture (rivers, roads, and coasts means trade routes and connections to your other religious cities, which typically means the religion will spread there quickly on its own).
The incans have the terrace which is the best early UB in the game as far as I'm concerned. It's the granary with +2 culture. Awesome to build/chop first in new cities.
After drama you can build culture in cities. Doing this for 3 or 4 turns at the start is a simple option.
Specialists can be used sometimes, particularly with some combination of caste system, mercantilism, sistine chapel, statue of liberty. Running an artist for a few turns pops your borders quick, at the expense of growth.