Italy never stamps my passport, neither on entry nor on exit. Maybe because it's an Italian passport? However, someone told me that it's because everything's electronic now, they just need to scan the passport. (In my case, they just look at it, and wave me on.) I just checked my Italian passport, and it has 40 blank pages, and not a single stamp. I've traveled to the US at least once a year, and often several times in the five years since I renewed my passport. (I use my US passport when entering and leaving the US, but otherwise use the Italian passport.)

I'm fairly certain that Italy also neglects to stamp non-EU passports, because I've seen complaints on TripAdvisor about the following situation: someone enters the EU in Italy, and their passport doesn't get stamped. A few weeks later, the person exits the EU at Schiphol airport in the Netherlands and gets a lot of grief because there's no entry stamp, meaning that it's not possible to determine if the person overstayed their allowed tourist visit time. All of the complaints I've seen have specifically involved exit from Schiphol airport, but it may be a problem at other airports.

In this day and age, it's hard to believe that the Dutch are that backward (or the Schengen countires) that they have no computer system set up to register people when they enter and leave countries. In fact, that contradicts what I've read about SIS, the information system they share which is maintained by France and updated every five minutes.
eg
https://www.rte.ie/news/europe/2017/...ecks-airports/

I think all of this is a non-issue, as I suggested initially, but, perhaps, more importantly, as others have suggested, because most places simply do not stamp passports anymore. Except if you ask for it as a souvenir. In Paris last year, I swear that the guy stamped it with a loud noise but there is nothing there. We have looked at our passports and, despite numerous trips to Europe in recent years, our only stamps have been in the Caribbean and Cuba. None from Europe. As bvlenci says, it is all electronic.

From: https://www.us-passport-service-guid...for-visas.html

>>A standard, 28-page U.S. passport book has 17 blank pages for visa stamps. The non-standard, 52-page book has 43 blank pages for visas. Most foreign countries require a specific number of blank visa pages in a passport as an entry requirement. Some airlines will not allow you to board if this requirement is not met.<<

My passport almost always gets stamped in Europe if I am entering or exiting the Schengen area. The only time it never gets stamped is when leaving the UK (it only gets stamped on the way in) and once last year, arriving in Rome the agent didn't stamp it and I had to ask him to. I would be nervous without a stamp, call me old fashioned but the last thing I want to do is get hassled by some dude for a missing one.

It's possible that the problem leaving the EU without an entry stamp is no longer an issue. I know I've read about it more than once in the Tripadvisor travel forums, but usually people reporting these things rarely say when it happened, I just did a search and found one such complaint from six years ago, but maybe the incident happened even earlier.

We fly twice a year from Boston to Nice entering Europe through Germany and exiting through Zurich. I can't remember the last time they have not stamped our passports both entering and exiting. Upon exiting, the officers have sometimes taken exaggerated deep breaths when they have trouble finding the entrance stamp. I'd say the computerization has a ways to go.