I believe travelling to Alpha Centauri at ~10 km/s would take of the order of 100 000 years (10 km/s is the order of speed of probes currently leaving the solar system). That seems 1. rather a long time to wait for a probe to arrive and 2. rather a difficult engineering task to produce a probe that will successfully operate for that long.
Various suggestions have been made for propulsion systems that would provide vastly faster speeds, but are hypothetical. Sling shot manoeuvres however are well established. Is there any limit to the velocity increase that can be achieved by repeatedly sling-shotting around a pair of bodies? I am assuming here that it is always possible to arrange the sling shot exit from one body so that it takes you on the correct course to sling shot around the second and then back to the first etc. (See https://en.wikipedia.org/wiki/Gravitational_slingshot for the principle of increasing speed by slingshotting).
Would it be possible to repeatedly sling shot to achieve a speed ~1000 times faster (~10000 km/s) so that the journey would "only" take 100 years?
My intuition says this will not be possible, because at high speeds you will need to get so close to each of the two bodies in order to redirect back to the other object that you will presumably crash into them. However, I don't know the maths and so wondering what the limit would be