What are the countries that have a year-round school system?

There aren't any countries with a completely year-round school system in the sense that school operates *every single day* of the year with no breaks. The concept of "year-round school" usually refers to systems that distribute breaks throughout the year rather than having one long summer break. Even in these systems, there are still significant scheduled breaks, just shorter and more frequent.

While many countries have varying school calendars, none operate entirely without scheduled breaks. The idea of year-round schooling is more of a scheduling model within a country's overall education system, with different implementations and varying degrees of continuity. Some countries might have schools that operate on a year-round schedule, but it's not a national mandate.

Therefore, the answer is none. No country has a year-round school system in the literal sense of no breaks.