If you only count planets, then I believe it's [MESSENGER][1] at six.

Though your question was explicitly: "Which probe that we have launched has received the most gravitational assists?"  The winner there is [Cassini][2], hands down.  It is on Titan flyby number 93, so far.  And it's flown by other moons of Saturn.  Plus the four planetary flybys on the way to Saturn.


  [1]: http://messenger.jhuapl.edu/the_mission/trajectory.html
  [2]: http://saturn.jpl.nasa.gov/mission/flybys/