In Pathfinder 2e, if you miss a night's sleep you become Fatigued, and you can get rid of this with a night's sleep.
But after this, I can't find any other consequences for missing even longer periods; can you just keep missing sleep forever and only remain "Fatigued"?
Note this is also interesting because this means you miss your daily preparations, presumably certain feats don't reset, like Hunt Prey.
There are no levels of exhaustion
Fatigue is a singular Condition to represent all levels of being tired. There are no higher levels or penalties for extended lack of rest.
You'll notice that it is not listed in the Groups of Conditions (Core Rulebook p619), only as its own Condition.