- The answer is: it depends!
- Are you in London or somewhere else with a large number of potentially suitable options? Otherwise, logistics may determine how many schools would be viable options, depending on whether you could give lifts for the next 5-7 years, and on the public transport and coach options.
- Do you want somewhere highly academic/moderately academic/gently selective? Single sex/co-ed/don't mind? Religious/non-faith? Large/small/traditional/modern/rural/city centre etc?
- How 'sold' on the idea of 11+ and independent school is your DC? If they are fully on board with it, it may be viable to consider more schools than if they are really resistant to the whole thing.
I would say 3 is a good number to aim for - one 'aspirational' school that you would love but may well not get as highly competitive/selective, one good one that you'd be happy with and have a decent chance of getting, and then a 'banker' - one you could all live with, and which you are highly likely to get.
That said in London it seems to be common to apply for 5 or so schools. Some schools may share a common entrance exam which then makes it easier to apply to more, though each school will likely have its own interview and other assessment procedures so there would still be more hoops to jump through. Again, depends how on board your DC is with the process.
We in fact applied to only 1 school for our DS as there was really only 1 which ticked all the boxes for us - largely because our son would hate a coach journey which meant we've had to go local. Our back-up is a local prep school for 2 years and then another 2 bites of the cherry to get in the school we really want.
But for our DD (who wouldn't mind a coach journey) we're applying to 3 - aspirational, our first choice (again our local school) and a 'banker' of a less academic one where we should get a space.
How to make it less stressful - we took the approach of starting them early on the preparation, from Easter of Y4, so we could be slow and steady rather than having a last minute panic to cram stuff in. We'd also talked to them from quite a young age about how the plan was for them to go to independent school, that most of their friends would be going to the local school, and that they'd need to take exams. So they didn't feel it was all being 'sprung' on them.
And then as PP says above, stressing to them that there are lots of great school options, that wherever they end up will be fine and they'll make lots of friends etc - not putting pressure on them in relation to any single school.