We spent a week in Rome this September, including a day in Vatican City. We saw fountains and pines, museums and monuments, art and history in overwhelming quality and quantity. But we did not see the Pope. Strangely, he never invited us to a private audience!
I don't consider that much of a loss, although we'd have been thrilled if he had, just as we would have been thrilled to be invited to visit almost any head of state. But seek him out? Never occurred to us.
The strange thing is how many people asked, after we returned, "Did you see the Pope?" When I visited England, no one asked me, "Did you see the Queen?" nor "Did you see the Emperor?" when we went to Japan.
It's a slightly more reasonable question with respect to the Pope, since we did go to St. Peter's Square, and the Pope has been known to address audiences there. But if he had done so when we were in Rome, we would most certainly not have been at the Vatican during that time, any more than we would be anywhere near Times Square on New Year's Eve. Not even for Pope Benedict—for whom I had a great deal of respect until he abdicated—would I brave such crowds. Certainly not for the man who might be a nice enough person but makes me rethink those exchanges that end with, "Is the Pope Catholic?"
On the other hand, Your Holiness, if you really want to meet with us, we'll be happy to consider a return visit to Rome. I still have an unused bus ticket as my version of throwing a coin into the Trevi Fountain.