My criterion for "visiting" a country is an overnight stay, and so this list doesn't include Guernsey or Pakistan, where I spent a few hours wandering around, but it does include the Vatican City. Yes, I actually stayed there, spending two nights in a hostel run by nuns inside the Vatican. The list also includes a few dependent territories (Aland Islands, Gibraltar, Macau, etc) and places which are far enough away from their homeland to feel different (Guadeloupe, Melilla, Svalbard, etc), a trio of unrecognised states (Artsakh, Northern Cyprus and Transnistria) and some places (Lundy and Seborga) which are fun to visit in a mouse-that-roared kind of way.
