I don't have access to the internet, but Wikipedia has many lists of useful information. You could find lists of monarchs for a few representative countries and estimate an average duration of their reign, then find a list of all monarchies that have existed, count them, and estimate an average age of each monarchical government. Then you can calculate an estimate by dividing the average life of a monarchical government by the average length of the reigns, and multiply that by the estimate for the number of monarchies there have ever been.
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