A traveling game/competition with the basic rule not allowing you to spend more than 24 hours in a single country, hence you have to frequently move from country to country. You also can't go back to countries you've already visited during the game, you can only move to the new ones. The aim is to visit as many countries as you can. If you break the rules you lose and can't continue.
I believe this would make an interesting show to watch, something to build a youtube channel or a TV show around.
Good "sport" for people who like to always be on the move and change places frequently.
An accelerated way to learn about geography and transportation from direct experience.
This would be similar to a strategic board game, but with real-life moves.
This could be done both as a self-challenge by a single or a few individuals and as a competition between individuals or groups of people. One person with a GoPro and some finances could start doing this and post videos on youtube to attract followers. If done as a competition, however, it would be more interesting to watch.
The organizers of the competition could offer effort-worthy incentives for the winners, like free flights on certain airline planes for the whole year, free hotel rooms from a certain hotel chain, etc. Those travel-related companies could also be the sponsors of the show, they'd get exposure this way. If the show was financially successful, the organizers could also give financial packages for the participants - you get a financial package of a certain value before the start of the competition and can spend it on transportation, accommodation, and food during the game.
Although one day seems like a catchy rule to abide by, you'd have to almost constantly rush to keep up with it and there would hardly be time for sightseeing. Different time periods could be given as alternative rules to abide by, 3 days seems like a good time for both moving in and out of the country in a less rushy way and having some spare time for sightseeing.
Another idea I came up with regarding this is to base the time you can spend in a country on its size - the bigger the country, the more time you have. The participants would have data tables with all the countries and the times they can spend there indicated.
Another factor that would be smart to include in calculating the time the participant can spend in a country is the difficulty of getting in and out of it. Some countries are very difficult to access either because of their location, the shortage of transport available, the difficulty of getting the visa or other required travel documents, etc. This would be especially relevant when aiming to visit all the countries in the world. One could come up with a formula for calculating an optimal time period for each of the world countries based on all the relevant factors.