The actor said the show had become an "extraordinary phenomenon” but had transformed his public profile to the extent that he was now seen as Logan Roy's "nice twin.”
Cox, who said he did not want to overstay his welcome in the role, said he was unable to go anywhere without Succession fans asking him to swear for them.
The Dundee-born star of stage and screen revealed that the creators of the show had an "end game” in sight for the warring Logan clan.
Speaking at the launch of his new autobiography in Scotland, Cox said the film industry and the Hollywood studio system had been left looking "old fashioned" by the success of long-form series like Succession.
Cox, who was born in 1946 in Dundee, made his name on stage in the UK, working with the Royal Shakespeare Copany and the Royal National Theatre in the 1980s and 1990s, when his first film roles included Braveheart, Rob Roy and playing Hannibal Lecktor in Manhunter.
More recent big-screen roles include Churchill, Planet of the Apes, The Bourne Identity, Troy and Zodiac, while Cox also had the lead role in the TV adaptation of Neil Forsyth’s Broughty Ferry-set Bob Servant books.
Succession, a comedy drama focusing on the dysfunctional dynasty running one of the world’s biggest media empires, was launched three years ago and sees Cox star alongside Jeremy Strong, Kieran Culkin and Sarah Snook as the three children at the heart of his empire.
In his new autobiography, Putting The Rabbit In The Hat, Cox insists Succession was originally sold to him as a “one-shot deal” which would see his character killed off at the end of the first season, only to discover during production that Logan Roy would survive.
In the book, Cox describes Succession as a “satire” about families, dynasties and entitlement.
He writes: “It’s about Logan Roy trying to teach his spoiled, entitled children the value of hard work. Teaching them the hard way. Teaching them in a way that is not always – hardly ever – moral or ethical, but teaching them nonetheless, because he wants the best for them, and because they are, at the end of the day, however spoiled and entitled they might be and whatever individual flaws they possess, his children, just as Waystar RoyCo is his child.”
However Cox insisted the show’s creator Jesse Armstrong and its others writers did not envisage it continuing indefinitely.
Speaking at the launch of his autobiography in St Andrews, Cox said: “I think some people, like Tony Blair, stay too long at the fair.
“I don’t believe in overstaying one’s welcome.
“Succession has been a slow boil. It’s now become an extraordinary phenomenon.
“I’m hell of a grateful to the show, of course, it’s been wonderful. But I have lost my anonymity. I’m no longer Brian Cox – I’m Logan Roy’s nice twin. So it’s hard.
“I know that the writers have an end game in sight. It’s just when they will play that end game.
“We are going to do season four, which we’ll start filming in June, and we’ll probably go to a fifth season, but I don’t think we’ll go any further than that.”
Succession won multiple awards for its first two seasons, including a Golden Globe for Cox’s portrayal as Logan Roy, while the first episodes in the third season have been broadcast in recent weeks to widespread critical acclaim.
Cox told the audience at his book launch: “We did the (season three) premiere in London. I’ve never known anything like it. The reaction was unreal. There were 2500 people at the Royal Festival Hall and it just went nuts.
"My wife thought it was like a rock concert. She said to me: ‘You’re a bit long in the tooth to be the leader of a boy band.’
“I can’t go anywhere without people asking me to do the obvious. I can’t say it here – but it starts with an ‘f’ and ends with an ‘f.’ Everybody wants me to do it.”
Cox recalled his astonishment at discovering towards making the first series of Succession that the writers had decided to make Logan Roy a Dundonian, a character he says he finds “constantly intriguing.”
Speaking in St Andrews, Cox said: “You can’t judge who you are playing. I’ve played some really horrible people. Some people find Logan horrible. I personally don’t. I think he is very misunderstood. You have to play them from their point of view.
“We know so little about Logan. But the thing about acting on screen is that you should always maintain a secret. It’s what propels you as the actor. You find it in the creative backstory of the character. But it’s the secret that gives the life to the flower that can blossom.”
Cox said the quality of new TV drama series was surpassing feature films because writers were being given more prominence and control.
He said: “The reason we’ve got shows like Succession now is that the writer was really given his or her place by the television industry, which they didn’t have under the film studio system. They were treated as workhorses.
"Writers moved into television and became producers in order to protect their own work. That’s why we’ve now got so much better television than we have films, which are now slightly old-fashioned.
"They discovered the long-form in television, which allows writers to tell a story in 10 episodes. There’s no longer a three-act structure. Now a story starts and it’s all about the development of the characters.
“From an actor’s point of view, when you’re working with a great director and a great show-runner on a show it’s just an absolute joy.”