There's a group for people who are well-traveled called the Travelers Century Club. It keeps a list of "countries" that includes some places that are not unique countries, but so out-of-the-way that visiting them counts. Here's my list (note, it's the TCC list -I'm not claiming that Alaska and Hawaii are countries):
Argentina, Alaska, Austria, Aruba, Bahamas, Balearic Islands, Barbados, Cayman Islands, Belgium, Guadeloupe & Dependencies, Belize, Hawaiian Islands, Bosnia and Herzegovina, Macau, Brazil, Martinique, Canada, Puerto Rico, Chile, St. Barts, China, St. Croix, Colombia, St. Thomas, Costa Rica, Scotland, Croatia, Wales, Denmark, Dominican Republic, El Salvador, Estonia, Finland, France, Germany, Greece, Grenada, Guatemala, Haiti, Honduras, Hong Kong, Hungary, Italy, Jamaica, Japan, South Korea, Liechtenstein, Luxembourg, Mexico, Monaco, Montenegro, Netherlands, Nicaragua, Norway, Panama, Peru, Portugal, Saint Lucia, Saint Vincent and the Grenadines, San Marino, Spain, Sweden, Switzerland, United Kingdom, United States, Vatican City, Venezuela, Vietnam
"Most exotic" is tough, because a lot of what constitutes exotic is timing. Does the former East Germany count as exotic? How about other countries visited when they were behind the Iron Curtain? China, back before Tienanmen? Vietnam in the 80s?