How do monarchs receive titles such as "... the Great"? I have tried to search for this but I cannot find anything informative. If you know anything about this or are able to find some information I will be very grateful!
A descriptive word following someone's name is not a title but an epithet. You may have better luck searching for famous monarchs/leaders with epithets.
Some examples off the top of my head:
Alfred the Great Ivan the Terrible Richard the Lionheart William the Conqueror Alexander the Great Edward the Elder Edward The Martyr Edward the Confessor Pompey Magnus
In some cases in ancient Rome people were awarded epithets. These had to be passed by the senate. Pompey was known as Pompei Magnus which basically means Pompey the Great.
In some cases they are self declared, in some cases they come to be known with an epithet after their lifetime in historical document, and sometime they are given an epithet simply to distinguish themselves from other monarchs of the same name which is basically the case for the Edwards listed above.