Most Americans who lack immigration privileges can only visit most other countries for up to 90 days maximum.
There are a few countries where Americans can stay visa free for up to six months (or longer depending on location):
- Canada
- Great Britain
- Mexico
- Panama
- Peru
- Bahamas
- Cayman Islands
- Several Carribbean Island nations including St Maartan, up to 1 year.
- Albania (12 months)
- All “U.S. Territories” including Guam, Puerto Rico, US Virgin Islands, etc – can stay as long as you want, no passport required.
- Svalbard – no visa required but very difficult to find housing, etc
- Georgia (the country, not the state) – 12 months
- Phillipines – can extend your visa multiple times up to 16 months
Additional countries and information here.