The role of a Scrum Master is to educate and coach the team into using Agile methodology and industry best practice. Scrum master will also train the team to work together to deliver on the team commitments and to be self-organizing. So, a successful scrum master will make the team into a fantastic self-organizing team that then don’t need a Scrum Master to facilitate/lead them.
However, in my experience, I have found that the role of the Scrum Master never goes away, but over time it reduces.
Scrum master needs to protect the team from external influences and needs to make sure team doesn’t become complacent. Also, I find that teams are never static for a long time and as new individuals join the team the Scrum Master need to make sure that the team stays gelled together. I have seen many times that one new member in a team can destroy the team atmosphere and happy working environment. Everyone is not used to working in the Agile environment, and they might find it hard. Every time there is a change in the team, job of Scrum Master become harder.
However, I find that Scrum Masters can take on more than one team but then cannot do multiple roles within the team. So, a Scrum master cannot be a product owner, and Scrum master for the same team or Someone cannot be a Scrum Master and developer in the same Scrum team.
In Summary, when starting a new project or a new team, Scrum Master should be solely focused on one team but has the team matures then Scrum Master will be able to take on another experienced Scrum Team.