The badges are for streaks on individual courses. Once your streak is more than a week long, your streak badges go up in increments of 5 days.
The streak clock counts down based on which time zone you have set in your settings.
The “best streak” is based on overall Memrise use, and the deadline each day is 12:00am GMT. So, you could do something in Course A on Mondays, Tuesdays, and Wednesdays, and then work on Course B on Thursdays, Fridays, Saturdays, and Sundays. Your streak badge wouldn’t increase, but your best streak would keep going up.
The “best streak” only goes to 365 days. (If it went higher, it would be longer than my streak badge for A2 German.)
Note: It is possible to interrupt/lose your “best streak” while keeping the streak in an individual course if your time zone setting is something other than GMT. For example, if you’re in the US, you could do Memrise in the morning on a Monday and in the evening on Tuesday, and completely miss the 24-hour period that was Tuesday in GMT.