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If the levels you add are of a class that's considered a "key class" for the monster's "role", then it's very simple: Existing CR + 1 CR per class level.
- Example: A CR 4 "combat role" monster with 4 fighter levels would add +4 CR for those levels, for a total CR of 8.
If the levels you add are of a class that's not considered a key class, then it's a little more complicated. You still start with the existing CR, but the first levels only add +1 CR per 2 levels (until the number of levels equals the original CR). Further levels after that add +1 CR per level.
- Example: A CR 4 "combat role" monster with 4 cleric levels would add +2 CR for those levels, for a total CR of 6. Adding one more cleric level would increase the CR to 7, since the number of levels was already equal to the original CR.
See the Monster Roles section for the roles of each Bestiary monster, and the Monster Advancement section (the part near the bottom) for a table of which classes are "key" for the different roles.