Personally, a best answer does not dissuade me from adding if I think there is possible value in my addition. Even when a best answer has already been given I will skim through the other answers to see what has already been covered and any discussion; chances are I looked at the question because the topic interested me or the questioner has helped me in past, or I am bored not just because the question is unanswered.
When to select a best answer depends greatly on the complexity of the question. In a simple question such as "When was X born?" An answer of "According to SOURCE a person named X was born on this day." deserves a best answer if it is for the right person. While other details and additional sources are awesome to add - the best answer can be awarded as soon as someone gives sourced details for the right person.
I am also of the personal opinion that best answers should only be used for answers of fact and solutions to problems, not to advocate opinions - that is what voting is for.