There is no rule. As for how to say it while still sounding like a normal human being, you would have to identify which/who is/are the necessary subject of the point you are trying to get across.
That is my brother's company's CEO's son's dog's toy.
The necessary subjects in the above sentences are: The brother, CEO, CEO's son and his dog. The company is not needed because it makes no difference to the point you are trying to get across. The focus of the statement is not on the company but rather, on the toy belonging to the son of a CEO.
That is a dog toy, belonging to the son of my brother's CEO.
You would have to accept one fact: Nothing belongs to a pet. Everything is allowed by it's master. So, in reality, the toy really belongs to the CEO's son, not the dog. Of course, many people would say "Oh, that belongs to my dog!" They are not wrong. However, in the context of your statement, the severity of touching the dog toy is not because it actually belongs to the dog, but to the son of the CEO of your brother.
Another statement from choster's answer,
I am your father's brother's nephew's cousin's former room mate
In this statement, I assume the father only has one brother because it does not state elder or younger.
A nephew is the son of a sibling. Your father's brother is called an uncle. And your uncle's nephew is your father's son which would be you! So, basically, the whole statement can be reduced to,
I am your cousin's former room mate.
From my understanding. the key to writing and/or speaking sound statements is rooted in identifying the key subjects of the point you are trying to get across. A CEO holds a good, or rather, a very strong weight that the word company is not needed in the context of your statement.