I went to 15 interviews, primarily because I just didn't have much of a clue where I wanted to go and wanted to see as many programs as I felt was feasible (for me). This turned out to be a good thing, since I only really felt like I could rank about 1/2 of the programs I went to and be happy. Given that I'm applying in general surgery, I feel pretty confident that I ranked more than enough programs.
Incidentally, I thought I had done some pretty good homework prior to applying and accepting invitations, but ended up being quite surprised at some of the programs I was impressed by and those I chose not to rank at all.