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Just had a quick look at the P&O brochure and all of their ships have Outside Cabins with 3rd and/or 4th person facility. They Outside Cabins with window, Outside Cabins with Balcony, and Suites.
Sorry I don't know about other cruise companies for certain, but I am sure someone else will be able to advise.
P&O certainly do and on some cruises the third person goes free. Just check and make sure the cabin has the type of bed you want. Many have bunk beds that fold down from the side or drop down out of the ceiling but some have sofa beds, notably Arcadia, where the sofa converts to a bed at night.
Most deck plans identify cabins with 3rd and 4th beds by using a * or ** or a small triangle (or something similar). For example, I have the Ocean Village brochure to hand and they have both fold down beds and sofa beds for 3rd bed occupancy depending on the grade of cabin. RCL also show cabins with sofa beds and pullman beds (whatever they are). Standard size cabins can get a bit cosy with 3 or 4 adults but you don't spend a lot of time in there and if you set up a rota for getting washed and dressed it's not such a big deal. Just book up and enjoy yourselves.
The simple answer is, all major cruise lines have 3/4 berth cabins.