Take a spring attached to an immovable point, and pull on it with a constant force f for a sufficiently long time t. Mildly counterintuitive fact: The stiffer the spring, the less work you do on the system.
Replace the spring with a damper, so that the system is no longer conservative, and your effort is essentially going to waste. Moderately counterintuitive fact: It's still true that the stronger the damping force, the less work you do.
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Counterintuitive consequence: In the limit, an infinitely strong damper is a constraint, and then the system is conservative.
