35 countries, and the Caribbean is where runs end
The United States, Canada, Mexico and Brazil come first, then most of South America. Central America is the first place the board slows: seven countries sit between Mexico and Colombia, Guatemala, Belize, Honduras, El Salvador, Nicaragua, Costa Rica and Panama, and most people can name four or five of them before they have to stop and think.
The real test is the Caribbean. The UN counts 13 sovereign states there, and only a few, Cuba, Jamaica, Haiti and the Dominican Republic, come easily. The small island states of the eastern Caribbean, Antigua and Barbuda, Saint Kitts and Nevis, Dominica, Saint Lucia, Saint Vincent and the Grenadines, Grenada and Barbados, are the stretch of the board that tends to stay blank longest.
Why North and South America share one quiz
The UN counts North America and South America as two continents, and so does this site: there are separate explainers for how many countries are in North America (23) and how many countries are in South America (12). The board keeps them apart too, with a live count under each heading, so you can read your result as two runs. They share a page because a recall quiz of 12 names is over in a minute, and because the Americas are easiest to learn as one long line from Canada to Chile.
Every country here is a sovereign UN member state. Greenland, Puerto Rico, Bermuda, the Falkland Islands and French Guiana are territories, not countries, so they are not on the board; type one and the quiz says what it is. Panama and Trinidad and Tobago are counted in North America, as the UN counts them, even though both sit right next to South America.
Tricks that make the 35 stick
Trace the land first. Start at Canada, come down through the United States and Mexico, cross the seven countries between Mexico and Colombia, then go round South America clockwise: Venezuela, Guyana, Suriname, Brazil, Uruguay, Argentina, Chile, Peru and Ecuador. Bolivia and Paraguay, the continent's two landlocked countries, are the ones to fill in afterwards.
Then do the Caribbean as an arc: the Bahamas in the north, the big islands of Cuba, Jamaica and Hispaniola (shared by Haiti and the Dominican Republic), then the chain of small islands curving south east to Trinidad and Tobago, with Barbados just off it to the east. The method is written out properly in how to remember every country in the world, and the countries and capitals of South America list pairs each of the 12 with its capital.