Can I leave a planet without achieving escape velocity? I know that if you exceed orbital velocity, you will never fall-back to the planet.  My question is not about orbits.  It's about brute-force propulsion to achieve altitude.  I'm using an intentionally slow velocity to help illustrate my point. 

Imagine I have a rocket with very efficient fuel storage.  My rocket can store enough energy to accelerate to 100kph shortly after leaving the ground, and continue to maintain that speed (100kph) for a very long period of time.

My rocket just goes straight up.  It doesn't try to enter an orbit.  As it leaves the atmosphere, it can throttle-back because there's no air resistance.  As it continues to gain altitude in inter-planetary space, it can throttle-back even more because Earth's gravitational influence diminishes with distance.  It just maintains enough throttle to continue moving away from Earth at 100kph.

At some point, Earth's gravitational influence would be moot, as other bodies (Jupiter, Sun), would gain relative influence.  Eventually, far outside the solar system, even the Sun's influence would be insignificant.

My rocket never achieved escape velocity, but it sure did escape.

Assuming that my fuel supply could last long enough, and I wasn't concerned about travel time, could this method allow my rocket to "leave" without achieving escape velocity?
The basic concept here, that you instead of relying on having a large enough velocity that the Earth can not pull you back in time just have a low and constant velocity, and keeps thrusting to counteract gravity instead, is not flawed by default, given your assumptions


  Assuming that my fuel supply could last long enough, and I wasn't
  concerned about travel time, could this method allow my rocket to
  "leave" without achieving escape velocity?


However escape velocity decreases by distance, so even though you are travelling at only 100kph, that is eventually going to be above escape velocity, as the escape velocity goes towards zero when the distance goes towards infinity. You can not really escape a gravitational field, so the only meaningful metric is if you have enough kinetic energy to not be pulled back. That is not possible by your constraint of not having escape velocity.

In a sense though, you can escape the Earth's gravitational field, using the common approximation of sphere of influence. It defines a region for Earth's gravitational influence, and ignores it outside of it. The border of the SOI can be reached without ever going at escape velocity.