Sword of the Stars had an interesting late game crisis in the form of locusts. These were synthetic life that formed large swarms, living in space ships that looked like a cell. They traveled from planet to planet, consuming all its resources. Then the "cell" would duplicate itself, and each duplicate would seek out a new world to consume. Worlds had a resource rating that could be depleted completely, making it essentially worthless. The cells had to be defeated in space combat.
I'm not sure how well this would translate into Stellaris, but the basic idea is pretty cool and could probably be integrated in some form.
A variation that plays more on the "resource depletion" aspect could be that the locusts are not represented as space ship but more akin to pests that spread like a wildfire and reduce the food, energy, and mineral output of planets and generally cause trouble on the infested planets. The idea is to destabilize an empire, reduce its income, and make ordinary challenges much more difficult.
This would only be interesting if the end result is that players are forced to adapt, the political landscape of the galaxy is changed, some weapons and technologies become less useful, things have to be done differently.
A plague variant could instead play on a deadly parasite that wipes out most of your population and causes loss of technological knowledge, communications, large economic losses, etc.
