How long can one travel around Europe with a US passport?
There is no limit for Europe as a whole (Europe is a continent, not a country). But:
For the 26 countries which are part of the Schengen Area., they collectively have a limit of 90 days in 180 days. The Schengen limit is much discussed on this forum, it is a rolling limit. If you exit it does not set the clock back to zero, it just stops the clock and it resumes when you re-enter.
Other countries have limits of 30 or 60 days.
State Department advice on passport and visa requirements for specific countries:
For example, Germany does not require a tourist visa for Americans who visit less than 90 days. Also see:
Germany is part of the Schengen Area and so the 90-day limit for all 26 Schengen countries includes Germany.
List of Schengen countries: Austria, Belgium, Czech Republic, Denmark, Estonia, Finland, France, Germany,Greece, Hungary, Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland
The US government link about the Schengen limits above is not correct. It says 3 months in 6 months. It is actually slightly less, 90 days in 180 days.
Note that Schengen and EU are different. England and Ireland are part of the EU, but are not Schengen. Norway and Switzerland are not part of the EU, but are Schengen.
If you want to stay in a country longer, you can get a long stay visa. But this takes some time and effort, and can't be done at the last minute. There are some posts here from people who have done this successfully; lots of documents are required (bank statements, proof of insurance, etc), as well as one or more trips to a consulate. Each country has different rules (whether they're in Schengen or not). So, if you're thinking of going this route, start researching now.