In addition to all of the above, I find adding a third dog to the family changes your interaction with your dogs more profoundly than adding a second dog.
Walking two dogs together is fairly simple even if they aren't the best trained dogs. You can use leash couplers that prevent the dogs from getting twisted up, you can more easily control them. If needed, your spouse/partner/child can take one dog so you can focus on the one dog. With three or more dogs, leashes get tangled (couplers don't work on three or more dogs), it is much more difficult to control them when you are being pulled in three or more directions, if you need to focus on one dog, you need to hand off two dogs to your spouse, partner, child.
Day trips, errands, etc. People tend to take one or two dogs with them when they do day trips or run errands, etc. But once you have three or more dogs, that changes and the dogs are left home far more often. Taking one, or even two dogs, out with you still allows you to focus on and enjoy the activity. With the third dog, it is more about handling the dogs then it is about enjoying lunch with friends, shopping, seeing the sights, etc.
I have had three dogs before, as well as one, two, and four dogs. For me, I find that more than two dogs at a time severely limits how much the dogs go out and do things with me. Instead, they tend to become homebound.
I enjoy camping, but many campgrounds have a two dog per campsite limit. So with three dogs, I have to choose which dog(s) goes with me and which get left behind. And that means I have to add in the cost of a pet sitter or boarding kennel every time I want to get away. This "two dog maximum" limit can be found at other places too - like hotels, parks, etc.
Friends and family are most always welcoming to one dog. With two dogs, that welcome mat gets yanked out from under my feet by some. With three dogs, I become the unwanted visitor.
If you never take your two dogs anywhere, then adding a third probably won't have much impact. But if you take your dogs with you a lot, then a third dog will most likely change that - a lot.
I've been lucky in that my dogs have gotten along well together regardless of genders. The one exception was when I had a female and two males - one of the males wouldn't let the other male go into the backyard. The other male hid out in the garage if I wasn't there to keep an eye on things. But, I have seen a lot more fighting issues with households of three or more dogs (regardless of sex) than I do with two dog households.
