Was talking about Voyager 2 with my friends, and we started wondering if the current tech would allow us to build a significantly faster deep space probe to study the heliosheath and beyond. Or would most of the velocity still be from gravity assist maneuvers, rendering any differences in propulsion tech meaningless?
It’s nothing built into the probe that is responsible for its speed. You’d need to launch it with a larger more powerful launch vehicle or as you said, design its flight to include planetary slingshots to boost its speed. Unmanned space probes don’t have significant propulsion of their own. Other than rockets for small course and to adjust their attitude, they’re drifting unpowered in their flight.
Not so these days. Some recent NASA probes, like the Deep Space probes, and Dawn, carry engines which are designed for long-term use. The ion propulsion systems in these craft don’t provide much power on an instantaneous basis, but they’re designed to fire for extended periods. In this article, it’s noted that the Dawn probe is expected to be using its engines for roughly 5 years out of its 8-year mission.
Ion drive would be useless for such a mission, then?
What about ion thrusters, don’t they theoretically have the ability to apply continuous acceleration for much greater periods than chemical thrusters (albeit with less force) and therefore allow much higher speeds over the long term?
Edit: beaten to the punch
If I’m understanding correctly, the probes use the ion engines to make significant course or speed changes over the course of the mission rather than just get them up to a higher speed at the start (to do that you can just have a booster stage that burns at a high thrust for several minutes then drops away when it’s burned out).
For example New Horizons, currently on its way to Pluto, had the greatest-ever launch speed for a man-made object, but all of its acceleration was due to a conventional rocket booster and a Jupiter gravity assist.
Dawn’s mission has it entering orbit of Vesta, then later this year it will accelerate out or Vesta’s orbit and proceed to Ceres where it’ll go into orbit.
Too soon for the inevitable Star Trek tangent, I suppose. 
Yeah–Voyager was faster, but only because (for whatever reason) it got a greater boost from the various gravitational assists than New Horizons did. That said, New Horizons could have gotten the same assists had the timing been right and ended up with a higher final velocity.
Also, there’s no reason the initial boost couldn’t be scaled further. The Saturn V could take 4x the mass to LEO as a the Atlas V that New Horizons used, this would have enabled a larger boost stage or even an entirely new stage. A smaller craft would also enable a greater final velocity.
It basically comes down to how much you’re willing to spend :-).
Nitpick, Voyager 1 is the current speed champion. Voyager 2 and New Horizons are going close to the same speed, about 2 km/sec less than V1.
Would you want to make a probe faster if the goal is to study the heliopause? Sure you get out there faster but you have less time to make your observations and relay the data. Not like a probe has any way to independently significantly slow themselves down once they reach deep space.
They can reorient themselves half way and begin decelerating, firing their engines in the opposite direction will slow them down at the same rate as they accelerated up to speed.
Since it’s taken years for the Voyagers to cross this zone, I don’t think transiting at a higher speed is necessarily going to be an issue.