I was tired of slow wars of attrition against the AI using just a small group of elite units- I wanted to make a massive army and crush them instantly. That's what this strategy is designed to do! This game demonstrates Civ V style infinite city sprawl (ICS) quite well. The settings are France/immortal/continents/standard speed and size, no ruins.
I was alone on my continent, so I could focus completely on development while ignoring my military. I settled about 6 cities, using the French special culture to take policies in liberty and commerce (cheap settlers, faster workers, 1 culture per city, extra gold in the capital, faster ships, and +3 hammers in all coastal cities).
Then, I stopped expanding, and just saved up all the culture and scientists that I could. I was lucky enough to be on a continent with wine, so I could build monastaries for massive culture. I also built the Hagia Sophia for faster great people, and the forbidden palace for 50% less city unhappiness. My tech target was steam power and railroad.
Around 1300 I reached that target, and entered the industrial era. I then spent all the culture I had saved up to head down through the order policies: faster buildings, 50% less city unhappiness (which made them free), 10% less building maintence, and +5 hammers per city. I had prebuilt a bunch of settlers (super cheap with liberty) and had them sitting in place, waiting for this. Now I was ready for some sprawl:

This was the result of packing in a city everywhere I could find a space. The AIs all told me how worried they were by my expansion after I did this.
each city produced 0 unhappiness except for its population, and a suprising amount of production. Here's what a coastal city looked like:

16.25 hammers and +7 food in a size 1 city- not bad. And as soon as that harbor finishes it'll get the railroad bonus (+50% production) too.
As a result, my production grew from 100 to 800 in about 35 turns, despite being last in population. 800 hammers/turn is a really huge number in civ 5, much higher than I've ever been able to get before. By the time this game ended I was up to almost 2000! In these turns, I also went from 0 military, to the largest in the world, and the most advanced as well.
I was producing at least 2 new foreign legion infantry every turn. The small cities still needed at least 10 turns to build one, but with so many that turned into a good rate of production. I had to stop after a while, because I had so many that was losing money despite being in a golden age. No problem, I just switched to making markets/banks and did some trading post spam:

Who needs anything else?
When the fighting happened, it was over very quickly. If you have the AI outnumbered and outteched, you can pretty much kill their whole army in 1 turn and grab all their cities the next. Mech infantry foreign legions are strong enough to kill cities in one blow.

This was about 1/3 of my army
If you're feeling frustrated by the low production yields and small army sizes of Civ V, I highly recommend this strategy. It's a little slow getting off the ground, but once you reach the industrial era it's unstoppable.
edit- My opinion about this strategy has changed since I originally wrote this. At first I was excited to win the game so convincingly, and to finally get the kind of production values that I wanted. Now I've realized that this strategy represents a huge flaw in the balance of the game. You shouldn't be able to massively outproduce deity AIs, or to completely ignore terrain when settling cities, and yet this strategy allows both those things. They need to completely rework the economic rules for this game, to give some sort of reward for having fewer and larger cities, and nerf the ICS strategy.
I was alone on my continent, so I could focus completely on development while ignoring my military. I settled about 6 cities, using the French special culture to take policies in liberty and commerce (cheap settlers, faster workers, 1 culture per city, extra gold in the capital, faster ships, and +3 hammers in all coastal cities).
Then, I stopped expanding, and just saved up all the culture and scientists that I could. I was lucky enough to be on a continent with wine, so I could build monastaries for massive culture. I also built the Hagia Sophia for faster great people, and the forbidden palace for 50% less city unhappiness. My tech target was steam power and railroad.
Around 1300 I reached that target, and entered the industrial era. I then spent all the culture I had saved up to head down through the order policies: faster buildings, 50% less city unhappiness (which made them free), 10% less building maintence, and +5 hammers per city. I had prebuilt a bunch of settlers (super cheap with liberty) and had them sitting in place, waiting for this. Now I was ready for some sprawl:
Spoiler :

This was the result of packing in a city everywhere I could find a space. The AIs all told me how worried they were by my expansion after I did this.
each city produced 0 unhappiness except for its population, and a suprising amount of production. Here's what a coastal city looked like:
Spoiler :

16.25 hammers and +7 food in a size 1 city- not bad. And as soon as that harbor finishes it'll get the railroad bonus (+50% production) too.
As a result, my production grew from 100 to 800 in about 35 turns, despite being last in population. 800 hammers/turn is a really huge number in civ 5, much higher than I've ever been able to get before. By the time this game ended I was up to almost 2000! In these turns, I also went from 0 military, to the largest in the world, and the most advanced as well.
Spoiler :

I was producing at least 2 new foreign legion infantry every turn. The small cities still needed at least 10 turns to build one, but with so many that turned into a good rate of production. I had to stop after a while, because I had so many that was losing money despite being in a golden age. No problem, I just switched to making markets/banks and did some trading post spam:
Spoiler :

Who needs anything else?
When the fighting happened, it was over very quickly. If you have the AI outnumbered and outteched, you can pretty much kill their whole army in 1 turn and grab all their cities the next. Mech infantry foreign legions are strong enough to kill cities in one blow.
Spoiler :

This was about 1/3 of my army
If you're feeling frustrated by the low production yields and small army sizes of Civ V, I highly recommend this strategy. It's a little slow getting off the ground, but once you reach the industrial era it's unstoppable.
edit- My opinion about this strategy has changed since I originally wrote this. At first I was excited to win the game so convincingly, and to finally get the kind of production values that I wanted. Now I've realized that this strategy represents a huge flaw in the balance of the game. You shouldn't be able to massively outproduce deity AIs, or to completely ignore terrain when settling cities, and yet this strategy allows both those things. They need to completely rework the economic rules for this game, to give some sort of reward for having fewer and larger cities, and nerf the ICS strategy.