Welcome to our Euro Venture!

You wanna go to Europe for two years? Not so fast.
The U.S. passport is blessed with a robust visa policy, but there are limitations. There is a large area encompassing most of Europe called "The Schengen Zone". The visa policy for the Schengen zone is that a U.S. citizen can spend 90 days out of every 180 days within the zone. After your first day in Europe, the clock starts, and 180 days after that, the clock restarts. While in the zone, you can move freely about the various countries, much like you would if you were traveling the U.S..

This is a bit of an issue for us because we want to travel for two years without returning to the U.S. for 90 days! For any other travelers without cats, the world is your oyster: Southeast Asia, Australia, etc. But aforementioned airport issues make it difficult for cats. Not to mention, our goal is to travel Europe, not the world (yet!).
There is a solution though, which is to bounce in and out of the Schengen zone, staying the allowed 90 days, then going to a non-Schengen zone for 90 days. Early on we found a resource from Nomadic Matt (https://www.nomadicmatt.com/travel-blogs/how-to-legally-stay-in-europe-for-more-than-90-days/) outlining this strategy. There are some notable changes though: Bulgaria and Romania have recently joined the Schengen zone as of March 2024! Also, Moldova has become a bit of a hot zone with the Russian-Ukrainian war. Nevertheless, the solution is the same, we just have less options.
One country Matt doesn't talk about is Turkey.
Turkey is a country rich in history and culture, Istanbul in particular. The food is great and there are tons of cats! Checkout my album from previous trips in 2017 and 2022: https://photos.app.goo.gl/921TPaFisSQ6Un647
For our initial travel itinerary, Istanbul is a reasonable (and scenic) drive from Budapest. Turkey is predominantly a Muslim country and there are areas where the U.S. State Department advises against traveling due to political tensions (visit https://travel.state.gov/content/travel/en/traveladvisories/traveladvisories/turkey-travel-advisory.html for the latest information). The West side of Istanbul has much more of a European influence and is generally safe for Western travelers. Another benefit to Turkey is that the U.S. dollar is strong, providing robust spending power that has been improving over the last five years.

Other great countries to visit are:
Ireland
Expensive, but a fantastic country to spend your non-Schengen days
The U.K.
Offers extended Visa of 180 days per year, meaning you could do all of your non-Schengen time there.
Cyprus
A beautiful island nation.
The Balkans
This includes Albania, Bosnia and Herzegovina, Kosovo, Montenegro, North Macedonia, and Serbia.
All viable countries which have less tourist traffic. Many of these countries experienced war and civil unrest in the 1980's - 2000's but have overcome those challenges and now make for excellent vacation spots. Give them a try - we plan on it!
Not harping on Matt here, but there is talk about bilateral agreements with countries to bypass the Schengen visa limitations. He goes on to admit that those agreements have been superceded by the Schengen agreement and to avoid them... We couldn't agree more, don't try it. Especially if you plan to remain in Europe, a violation could get you deported with a ban on re-entry.
In conclusion, we can see that spending two years in Europe isn't easy, especially carting around a couple of felines. However, the strategy outlined here could allow you to travel the region indefinitely.