For private grammar schools your child can sit as many 11+ exams as you wish. For state-funded grammar schools, your child will have to sit the exam in their primary school. If your primary school is not in a Local Education Authority then you will be able to sit multiple 11+ exams at selective schools.
When listing your preferred schools on the application form, make sure to fill in all the possible options and don’t list only one school. You might end up in a situation in which you won’t be able to get into your preferred school, so it is important to have thought of a plan B and C. Performing well on the 11+ exam is not equal to and certainly not a guarantee for getting accepted by your preferred school. So do take the preference form seriously and utilise all the possible options to avoid the scenario in which you only list one school and don't get offered a place.
Why not give the 11+ courses a go? They're free!