What happens if your passport expires while you are visiting another country?

It entirely depends on the country.

For the UK, there is the following guidance issued for the public:

You’re from an EEA country or Switzerland

You can enter the UK with either a valid passport or a national identity card issued by a EEA country. It must be valid for the whole of your stay.

You’re not from an EEA country

You must have a valid passport to enter the UK. It must be valid for the whole of your stay.

If you try to enter the UK on a passport which will expire during your intended visit, one of several things may happen:

• you may be denied boarding by the airline or ferry company
• you may be denied entry at the border by the Immigration Official
• you may be allowed entry for a shorter period of time

The United States has the following guidance for the public:

Visitors traveling to the U.S. are required to be in possession of passports that are valid for six months beyond the period of their intended stay in the U.S.

However, they also publish a list of countries which are exempt from this six month rule, called the "Six-Month Club".

Citizens of the countries listed below are exempt the six-month rule and need only have a passport valid for their intended period of stay.

Six-Month Club

Other countries will have their own approach to this issue.

Generally speaking, the period you are allowed to stay in a foreign country is shorter than the time you have until your passport expires. So, your question is almost purely hypothetical.

If your passport does expire while you're abroad, and your passport is the only valid travel document that's accepted for your nationality in that country, you're probably breaking some other law as well, like overstaying your visa. But, no one can smell your passport expired, so, if your passport doesn't get inspected in-country, and it's not inspected thoroughly when leaving the country, you could just get away with it.

Otherwise, you can go through your embassy or consulate, or the embassy that represents your country, to obtain a temporary or emergency travel document. Though, you still might up ending being fined by the host country.

Usually you wouldn't be allowed to travel if there's less than 6 months left on your passport anyway, to prevent this issue.

If however you say, had your trip out delayed, or some other unforseen situation (illness?), it would depend on the country, and your best bet would be to contact your embassy. At that point it'd likely be similar to losing a passport - you're now out of your home country with no valid travel documents, and so you could get an Emergency Travel Document issued at best to get you home. (British Example).