After the first answer, each subsequent answer is less likely than the previous one:
The above graph shows the distribution of open questions by number of answers. Not surprisingly, people have a strong tendency to answer unanswered questions. But once the question has been answered the odds of one more answer decrease somewhat. This looks roughly like a Poisson distribution with a mean of 1.5 (the answer rate on Mathematics):
Outside of the relative lack of questions with zero answers, answers are more or less distributed as we might expect from a Poisson process. If so, the odds of a new answer are independent of the number of existing answers. As a first approximation, I say that getting one answer does not artificially discourage further answers.
Another way to think of it is to consider what we'd like to avoid. It would be annoying if a person with great insight into a questions decided the existing answer was "good enough" and the site were deprived of a better solution. But remember that they have other options such as editing or leaving a comment (which sometimes prompts an edit). So even if you never get another answer posted, that doesn't mean you'll never get a better answer.
We know there is a first answer bias. On Math, it's about 0.6 votes on average. Some of that might come from the bandwagon effect. But it could also be that the initial answer legitimately tends to be better than the rest. This site boasts excellent quality, so it stands to reason that a quick answer that's upvoted by the community sets a high standard. The next answer might very well fill in some details or provide an alternate method of looking at the question. Those are useful things and we hope people offer them as answers too, but not all voters will find them as worthwhile as the initial answer.