In a set of N people, how many people share birthdays with someone else in the set? The famous birthday problem runs as follows: in a set of $n$ randomly chosen people, what is the probability that some pair of them share a birthday? The following question is an extension of that famous problem.
Given a set of $n$ randomly chosen people, there is a probability that $k$ people in the set share a birthday with someone else in the set. For a given $n$, what is the most likely $k$? Also, for a given $n$, what is the most likely configuration of birthday sharing, i.e. how many pairs, triplets, etc.?
Note: I am assuming that there are 366 birthdays in a year and that each birthday is equally likely.
 A: Letting $m=$ number of days, $n=$ number of people, $k=$ number of people with shared birthdays. Then $j=n-k=$ number of "singletons". 
The problem is equivalent to the following urn-and-balls problem: place randomly $n$ balls uniformly inside $m$ urns, find $P(j)$ , distribution of the number of single occupancy urns (singletons).
A simple (perhaps too simple) Poissonization approximation gives
$$E[j]\approx n\, e^{-n/m} \tag{1}$$
We can expect that asymptotically $j$ tends to a normal, so that the most probable value is near the mean. 
A graph for the average fraction of shared birthdays ($1-E[j]/n$), under the above approximation and for $m=366$ is shown below. It agrees quite well with the graph from Marko Riedel's answer.

This paper studies some asymptotic approximations in more detail.

Edit
Actually, there's no need to do a Poissonization approximation to compute the mean. The probability of a given urn of being a singleton is
$ \frac{n}{m} (1-\frac{1}{m})^{n-1} $, hence 
$$ E(j)=n \left(1-\frac{1}{m}\right)^{n-1} \tag{2}$$
Obviously, $(1)$ and $(2)$ are asympotically equivalent for large $m,n$.
