Large countries, as well as countries with overseas dependencies, experience on average less than 12 hours of night each day.
I calculated, or rather measured, daylength for each country in the world (using Natural Earth Data and a script created from this example by mbostock) including their dependencies. Since exposure is different each day of the year, the result is an average of this (Norway or Antarctica among others would have no hours of night in their respective summers, although both would have some hours of day during their winters).
The result shows two countries on which the sun never sets, France and the UK. Antarctica is close thanks to the few pieces of land that lie outside the polar circle. The United States dependencies in the Pacific also afford them a large amount of daylight, as well as Russia's sheer size (spanning almost 180°, a large part of it also being north of the polar circle).
Overseas territories are highlighted to give a better idea of the span of countries.