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Well-known physics establishes that, compared to some reference frame (I'll denote as "stationary" here), one can do several things to start at rest, end at rest, and experience less time than objects in the stationary reference frame do, although one can not experience more time given the start and end at rest condition. The former can be accomplished by the following:

  1. Take a trip at relativistic speeds
  2. Approach to an object in a deep gravity well, approaching an event horizon potential

Question: Can either of these be done for an object with mass $m$ while expending a small amount of energy to do it, formally using an amount of energy $E\ll m c^2$? To be sufficiently clear, the requirement for traveling to the "distant future" is satisfied if $\Delta t_m \ll \Delta t$, meaning that the time experienced for the object in question is much less than that experienced by the stationary reference frame.

Conventional thinking is that #1 is simply not possible given my requirement. Any physical ship that let you planet-hop while living a single human lifetime would require spaceships that probably expend much more energy in the propellent than what the rest mass of the spaceship is. This is the relativistic rocket problem.

Thinking about #2 is what caused me to ask this question. Imagine a large black hole and a spaceship that starts off at a great distance from it. It then uses a small amount of energy to enter into a very highly elliptical orbit that comes close to the event horizon. I think (although I am not sure) it would seem to the stationary reference frame that the flyby takes a small amount of time. The spaceship would be going very fast (large time dilatation), but only for a short time and thus could not travel to the distant future and make it back out without using huge amounts of energy. You can introduce other possibilities, like a stationary planet-like surface where a traveler could sit to pass the time, but the escape potential renders this useless too.

Is it a fundamental reality that we can not use time dilation to go to the distance future without using extraordinary energy to do so, or do I simply lack the creativity to think of how?

Alan Rominger
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One problem with the black hole flyby solution is that stable orbits are bounded away from the event horizon - any object free-falling through the photon sphere will wind up passing through the event horizon. If you don't feel like using a lot of propulsion energy, this bounds the amount of time you can spend in each pass where your time is highly dilated relative to a distant observer. On the other hand, if the black hole is rotating quickly, you can extract energy for your next flyby using the Penrose process.

Scott Carnahan
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Well, as is mentioned here, time dilation is caused by gravitational potentials, but not necessarily gravitational forces. So, it might be possible to travel between two equally sized black holes to experience extreme time dilation effects. Traveling through would not cost much energy, however it would potentially have an extreme effect on the time dilation.

Note however that I have never taken a GR class. I only noticed this from the other question and thought it would be relevant here. The equation given by one of the answers there is incorrect because you are no longer in the weak field limit, but I don't know what the strong field limit is.

I'm making this community wiki so that someone else more knowledgeable than me can clarify.

Community
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Andrew Spott
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Put the traveler in a bucket and lower him close to the event horizon of a black hole of mass $M$. Counterbalance him with a weight, which stores the potential energy he loses. Keep him there for a gazillion years, then raise him back up, releasing the energy stored in the counterbalance.

For an observer who can tolerate a given gravitational acceleration, you have to make $M$ sufficiently large.

The rope has to be strong enough to hold the bucket. This is a problem in practice but not in principle. The only fundamental limitation on the tensile strength of a rope is a relativistic one that basically says that the speed of sound in the rope can't be $>c$. This limitation is equivalent to limiting the tensile strength of the rope so that it can't be lowered past the event horizon of a black hole without breaking. But we don't need to lower it past the event horizon, just arbitrarily close to it.