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MANTIS was in 2015 a proposed Discovery class mission which were to fly by seven different asteroids including at least one NEA. I think the spacecraft which holds the record in number of flyby's is Voyager 2 which flew by four planets. Also New Horizons flew by an asteroid, Jupiter and Pluto and is heading towards a KBO. There have been spacecrafts which have done many flyby's of Earth and Venus for gravity assists, but asteroids are in different orbits and do not offer gravity assist.

What kind of trajectory or tour would allow for as many as 7 flyby's? Is it easy to pick and choose asteroids of interesting types and families, or would only a few alignments be available each year?

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    $\begingroup$ This Asterank might prove useful in understanding the concept of multiple asteroid flybys. $\endgroup$ – TildalWave Dec 26 '15 at 22:23
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By early 2014, we had discovered 630,000 asteroids (Youtube video visualizing asteroid discoveries) in the main asteroid belt.

Asteroid belt

The MANTIS team has found:

Many trajectories exist that provide required diversity

but I haven't found more information on trajectories (or much of anything beyond the two PDFs linked by Localfluff and me).

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  • $\begingroup$ You might add a reference to the LUCY mission. There are a few links here and in answers there as well. $\endgroup$ – uhoh Mar 4 '19 at 16:39
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Since my comment wasn't heeded I'll leave this as a separate answer. From information in the question Where is Lucy going? (asteroid mission) we can see that LUCY is doing just this. Note that the flyby's of asteroids involve way less gravitational effects, so it's not like flying by seven planets, these will be at safe distances (in case there is a debris field near any of them), that plus the low masses of the asteroids means that if corrections are necessary after each flyby, they will be quite small.

The first orbit below looks weird because it is in a rotating frame, in this case rotating with Jupiter (at the top). If you look at the second image, or watch the video http://lucy.swri.edu//mission/Tour.html) from this answer you can see that it's just an elliptical orbit that passes Jupiter's orbit twice. As explained in that answer, it turns out that for this particular periapsis and apoapsis, things in Jupiter's orbit move about 120 degrees, so you sample the leading set of trojans the first time and the trailing set the second time.

See also How well the orbits of Jupiter Trojans to be visited by Lucy may be predicted?

LUCY's orbit visiting Jupiter's Trojan asteroids

LUCY's orbit visiting Jupiter's Trojan asteroids

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