@DescheleSchilder honestly I have no idea whether you could get fast enough to cross it, hit perigee and come back out again. You couldn't end up in it though - you'd have to be travelling at the speed of light to do that I think (my understanding is the photonsphere is composed entirely of photons in circular orbits). The time dilation would be enormous if you did go through it - you'd be travelling very close to c AND very deep in a gravity well. Maybe it would make a good question to post? "Is it theoretically possible to slingshot round a black hole and cross the photonsphere?"

@DescheleSchilder if there weren't any further complications, yes, I believe it could. However you would need two orbiting black holes to achieve that, so definitely in the realms of theory here, at least as far as our own solar system is concerned. One complication that would arise as you get closer to c - the slingshot maths assumes that the mass of the object that is doing the slingshot is much smaller than the object you are sling shotting around.. As you get faster, your mass starts to increase, and as you approach c it tends to infinity i.e. more than the black hole! So another limit.

@BrendanLuke15 apologies you are right the v and u in the diagram are the opposite way round to elsewhere. It is possible, because the next planet you are going to is closer than infinity. The key thing I learned in order to get the answer to my question is that the voyager et.c slingshots were above escape velocity and so followed a hyperbolic trajectory, wheras if you travel close to escape velocity you can go into a parabolic trajectory or even a highly elliptical orbit, which takes you back to the other planet. However this does limit the maximum speed you can achieve, which is the answer.

@BrendanLuke15 PS just in case I have misunderstood what you were asking, if the orbitial velocity itself was 0 - there would be no point in trying to slingshot, round it... but then if u was zero, the planet wouldn't be orbiting, it would be about to start falling into the sun I think?

@BrendanLuke15 great diagram! From the frame of reference of the planet (top half of the diagram) you have gained nothing... however the planet is not the frame of reference we are interested in since we don't stay with it, we leave the planet behind. What we have really gained is shown in the bottom half of the diagram. This is what real slingshot manoeuvres are based on, not just of our own spacecraft, but also how we calculate the future trajectories of near earth asteroids.. E.g. Famously 99942 Apophis will slingshot past Earth in 2029 - en.wikipedia.org/wiki/99942_Apophis Cheers

@BrendanLuke15 great question - this is the key. From the intertial frame of reference of the planet - you lose all of it i.e. it takes back everything it gave you on the way in. However from the frame of reference of the sun (which it is orbiting at velocity u) you gain up to 2u. See en.wikipedia.org/wiki/Gravitational_slingshot in particular this diagram: en.wikipedia.org/wiki/Gravity_assist#/media/File:GravAssis.g‌​if Cheers

@Peter-ReinstateMonica It's a good point. I agree that three planets (suitably placed) would allow you to keep cycling round the entre system, and gain speed every on every slingshot. However, 2 planets is actually enough, because of the following two effects. Firstly, you can approach the inner planet roughly at a tangent, so maximum gain from that planet (i.e. more or less a 180 turn around, in the direction of orbit), whereas the corresponding loss from the outer planet will be smaller because you approach it at more of a right-angle. Secondly, the inner planet has a higher orbital velocity

@Peter-ReinstateMonica on your point about needing mega-Earths, certainly a dense rocky planet would give a better max speed (surface escape velocity) than a gas giant of the same mass... however in order to ping-pong you only need that the planetary escape velocity exceeds the solar escape velocity. Once you get out as far as Saturn, the solar escape velocity is around 14 km/s, which is far overpowered by Saturn, which has a surface escape velocity of 36 km/s, allowing you to swing all the way round the planet and back again (or even drop into orbit) e.g. at 20 km/s - way above solar escape

@DescheleSchilder yes you stay outside, just the same as when you gravity slingshot around a planet - the only difference with slingshotting round a black hole is it is much denser and therefore smaller, so you can get closer, which means you can go faster :-)

@RossPresser you are right that it will decrease as speed increases if you maintain a fixed distance from the object each time you pass it. However if you move closer to the object each time you pass it, so that you exactly compensate for the speed increase, you will get the same deflection. This is borne out by the escape velocity formula which is inversely proportional to the distance... which I was not aware of when I posted the question, but thanks to you fine people I have now come across :-) en.wikipedia.org/wiki/Escape_velocity