Extended Visa

The short answer is that it is often possible but it can be a laborious process. Google for people's experiences and go to the website of the Italian embassy (consular services) in your country for official statements.

Based on information posted by others and provided by an Australian couple I met who had successfully completed the process, you can expect to need at least the following:

• Proof of health insurance
• Proof of fairly substantial financial resources (income and/or assets)
• Lease for the place you will be staying in Italy. Not sure whether this can be finessed if you just want to travel around.

The Australian couple said they made three or four (I don't remember which) trips to the Italian consulate before they got everything taken care of.

You might use the search feature here for prior discussion. There are several postings from people who have obtained extended visas here. Generally time consumer and lengthy. A lot depends on exact where you want to go. It would be difficult to just travel for a year.

This is one website you can read through to start with and it will give you some ideas: https://www.internations.org/italy-expats/guide/moving-to-italy-15325/visa-requirements-for-italy-2

Go directly to the official site of the Italian Consulate, which will give you all of the details you'll need. I wouldn't waste time going around in circles on travel or ex-pat forums, which often have out-dated information.

It is possible to spend a year in Europe, with the proper visa, though you will need to meet several requirements, including being able to support yourselves during the time you're there.

If you expect to apply for a "talented or artistic" visa, be aware that this takes more time than it's worth.

each person relocating including children will need a visa and do the paperwork

Which country? What nationality are you? Are you intending to work (work visa) or just stay paid for from funds you already have?

You mentioned Lucca, which has led everybody to guess Italy. But, if you want a visa to stay past the "90 days in any 180" limit for tourists in the Schengen Area which applies for tourists from some countries, that implies you want a residence visa, which is handled separately for each country, and different rules apply depending on your nationality.
A residence visa for one country is only valid for that country. If it is a Schengen Area country that normally allows visits to other Schengen countries, but the "90 days in any 180" limit still applies for those other countries.

I would address your ability to travel to other Schengen countries and to enter and leave the Schengen zone multiple times to the embassy you are applying to.

All are conditions of the visa and can vary. This may even include limiting you to that country, a few, or all countries. I do not believe the 90 in 180 rule would apply, unless it is a condition of the visa, but that would be onerous to track, if even possible.

Paul, regarding the "90 days in any 180" limit if you have a visa from one Schengen country, it is, as you say difficult to enforce as there are no checks travelling between Schengen countries. But, AFAIK, a visa for one country does not entity you to stay above the limit in other countries. If you no longer have an address in the country of your visa, or were caught living (or even worse working) in a different country, the authorities are likely to expel you.

Do not expect to get a visa from one country and be able to spend as long as you want in other Schengen countries.

Regarding long-term travel plans assuming standard visa waiver rules for an American, can anyone provide their experience, or offer potential itineraries for traveling through the UE, UK, Ireland and Scandinavia for more than 12 months without violating the 90 in 180 day Schengen rule? For example, I have heard that Americans can stay up to 90 days in the Schengen area, then travel to the UK for 90-180 days then back to Schengen for another 90 days then back to the UK and so on and so on indefinitely. Does this make any sense?

Also, I have read that there are certain bilateral agreements between countries that supercede the Schengen rules. The U.S. apparently has such an agreement with Denmark which allows an American to extend their Schengen trip up to 90 days beyond the standard 90 days if he/she travels to Denmark and returns to the States from Denmark without traveling through another Schengen country. Does anyone know of other such agreements with the US?

From the New To Denmark website:
"Citizens of Australia, Canada, Chile, Israel, Japan, Malaysia, New Zealand, Singapore, South Korea and the US can freely enter and stay in Denmark for up to 90 days in any 180-day period, regardless of whether they have stayed in another Schengen country prior to entry into Denmark. The 90 days are counted from the entry date into Denmark or another Nordic country. If you have previously spent time in Denmark or another Nordic country within the previous 180 days, that time will be deducted from the 90-day maximum."

I'm no immigration lawyer, but I agree with your interpretation of the Danish policy.

As for the general rule, and remembering that the Schengen count includes both day of arrival and day of departure, it's my understanding that you can do this:

90 days in Schengen area
90 full days outside Schengen area
(Here ends your first 180-day period)
90 days in Schengen area
(Now you are adding one new Schengen day and dropping one old Schengen day as you progress, never exceeding 90 days)
90 full days outside the Schengen area
And repeat

However, Immigration officials often have the authority to make their own decisions, and I don't know whether such a pattern would raise eyebrows.

The only point I will reinforce is that entry to a country is always at the discretion of the authorities at the border. This is less of an issue for the Schengen countries as long as you honor the 90/180 rule, more of the issue for the UK where there is no "guaranteed" stay, just "up to 6 months", and as was noted, if they see a pattern, you may be denied entry.

But, theoretically, as long as you alternate between Schengen/non Schengen countries, you can travel indefinitely.