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Is there any specific, practical reason why we couldn't have a probe swing by Saturn, Uranus, Jupiter multiple times, over and over, before getting hurled into interstellar space at insane speed, overtaking Voyager and reaching neighbor solar systems within a reasonable timeframe?

From what I understand, while probes use their fuel for accelerating during gravitational loops, that's not essential - they could just use minimal amount of it to adjust route, to enter and leave the slingshot at optimal angle, and repeat it pretty much indefinitely, with a significant speed gain each time.

Is there any specific reason we don't have some probe accelerating to speeds that would leave Voyagers far behind, currently? What are the practical limitations of using this method repeatedly to bring probes to extreme speeds?

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The easiest explanation would be that eventually you'd gain enough momentum to reach escape velocity, at which point you'd actually have to decelerate to return to orbit from what would be a hyperbolic trajectory describing your maximum delta-v gained from an Oberth maneuver (powered flyby). With unpowered flyby, your theoretical maximum delta-v gain from a slingshot would be the body's own orbital velocity, there simply isn't anything more to be gained from it in terms of accelerating the spacecraft.

It then doesn't matter if you can achieve that during your first flyby, or do a flyby past another body to later return back to the same one many times over. You could only increase efficiency of an unpowered flyby up to 100% of the momentum that the gravity assist body can lend to your spacecraft, which is your initial velocity, plus body's orbital velocity. And with Oberth maneuvers, you'd be consuming your propellants with each flyby.

There actually exist such interchanging orbits between two celestial bodies, for example the Aldrin Cyclers (here's a YouTube video of one such example). And the reason they work is because of the conservation of angular momentum, which is another way of explaining why multiple such flybys you describe wouldn't work. Well, they would work, but not in the way of gaining some additional momentum out of nothing.

The optimal trajectory for leaving the Solar System is known as the Krafft von Ehricke trajectory.

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