VisitedPlaces.com is a free travel map maker. Click the countries you've been to and they fill in, while the counter keeps score: how many countries you've visited, and what share of the world that is. Think of it as an online scratch map, without the scratching.
Switch to the US states, Europe, another continent, or the regions of a single country from the Map menu.
Is it free? Yes. There's no signup and no account: your map is saved in its link, so keep the link.
How do I count the countries I've been to? Mark them on the map. The counter at the bottom and the chart in the corner show how many you've visited out of the total, and the percentage.
Can I show where I've lived, or where I want to go? Put those places in their own step, with a different color and label.