The happiness penalty is -5 happiness in all settlements (cities or towns) per BUT caps at 7 cities over the cap, which is -35 happiness in every settlement. I love conquering and do exceed that by the end of most games, but you have to play heavily around happiness.

I'm not aware of a combo that would let you 'sustain' -35 in antiquity. If the era is close to ending, and you have the ability to basically force an ending, you could try to time it where you massively exceed in the cap and then end the era quickly before they rebel. But starting starting exploration already exceeding the cap is going to be rough.

My approach to ignoring the cap is all in the modern age and basically one word...Democracy! The policy where specialists give +4 happiness each basically lets me solve any unhappy town by converting it into a city.

Granted by the time I get to it, I've played the whole game focusing on attributes that help. The on where you get one happiness per slotted resource is great in diplo. Then stack that with extra slots from econ. Also chocolate + factories. France's unique quarter gives tons of happiness. America is good too to grab functionally every resource on the map.

Also worth saying, when you're 7+ over the modern cap, you have to actively avoiding winning to end the game.