I have heard of football fans giving their children the names of the 1966 World Cup winning team as middle names. I'm sure one case I heard of was a girl as well!
So I don't think there is a limit but most people stop at two middle names.
"Tea is still believed, by English people of all classes, to have miraculous properties. Whatever your mental or physical state, what you need is 'a nice cup of tea'." Kate Fox, Watching The English.