I understand with current technology we can't just fly a straight line out of the solar system but which way out would need the least fuel?
Currently to navigate the solar system it is a dance around the planets using sling shots, gravity assist and such.
The Ground Tour trajectory of Voyager 2 "used" the first three of the four large planets Jupiter, Saturn, Uranus, and Neptune, but this was optimized for time, and it ended up with far more than heliocentric escape velocity.
Suppose instead the goal of the exercise were to barely achieve heliocentric escape velocity using the minimum fuel or delta-v, starting from LEO, with a lot more flexibility on time, say roughly 100 years from launch to achieving escape velocity (C3=0). Assume you can start at optimum configuration of the planets within their orbits.
What would that trajectory look like? Would it still use all four of these planets, or could you get by with fewer? Would it make sense to look inward, using the four rocky planets instead?
As a side question, would having even more planets present on the way out always help?
above: Voyager 2 Grand Tour. Source
below: Voyager 2 Grand Tour Heliocentric velocity. Source