On another question - Is an Immune creature considered to have the condition without suffering its effects? - it was concluded that a creature could gain exhaustion levels, while being immune to the effects of those levels.
Assume we have a druid/barbarian (Berserker subclass).
The druid hasn't been sleeping for a few nights, and so has 2 levels of exhaustion. There's a blizzard going on, bestowing another level of exhaustion.
He goes into a frenzied rage, taking the exhaustion level to 4.
He goes into the Wild Shape of an elemental, gaining immunity the effects of exhaustion. He then sprints enough times to the point where each Dash increases the exhaustion level by 1.
The rules aren't exactly clear whether exhaustion immunity prevents the exhaustion level from increasing, but regardless, he won't suffer the effects of exhaustion.
If he dashes enough to increase the exhaustion level to 6, what happens if he gains another level of exhaustion? Is it possible for exhaustion level to increase past 6?
If so, would this increase the amount of healing/spells/potions that remove exhaustion levels needed to decrease the exhaustion level to a point the Druid could drop Wild Shape without dying?