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?