[Varyon]

The Misttorn Witches book series had an interesting option - the country is divided into Princedoms, ruled over by a king (I think the term used is High Prince or similar) who is also the Prince of one of the Princedoms. Every so many years (I think it was 7 in that series), the Princes all come together to elect a new king from their number. A good deal of the series’ plot revolves around two sub-Princes trying to prove themselves worthy successors for their ailing father to pass down his title to in time to have a chance at becoming king at the next election.

For my own entry, perhaps a Herocracy. This could be similar to the point system previously suggested, but only performance on the battlefield counts - you’d get points for each battle you participate in, points at certain kill-count thresholds, points for achieving important objectives (like killing/capturing an enemy commander), points for particular acts of heroism, etc. Whenever the current king dies or chooses to step down, whoever has the most decorated military career becomes the new king. My own head-canon for such a setting is that the citizens are questioning the system after the shockingly-effective rule of Harold the Porter King - a man who served (as a porter) in a huge number of battles and ended up king after the previous king and all the generation’s “heroes” died in a disastrous battle.