Sandia National Labs and Northrup Grumman have been working on nuclear drones that could stay up for months at a time. NASA has even been given a proposal for a nuclear powered drone mission to Titan.
(Dragonfly nuclear drone mission to Titan)
The idea is that a nuclear-powered drone could not only stay up for a very long time, but could carry more payload, varied weapons and stay on station for longer.
As I recall, the eyes, world-spanning drones from Roger Zelazny's 1966 story This Moment of Storm must have used some sort of very long lasting power source, since they stayed up almost indefinitely. Just the thing for one person to use to monitor an entire planet.
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Humanoid Robots Building Humanoid Robots
''Pardon me, Struthers,' he broke in suddenly... 'haven't you a section of the factory where only robot labor is employed?''
Stratospheric Solar Geoengineering From Harvard
'Pina2bo would have to operate full blast for many years to put as much SO2 into the stratosphere as its namesake had done in a few minutes.'