Is it theoretically possible to escape a classical black hole with force? A stone being thrown from the surface must have a velocity of $11.2$ km/sec if it wants to escape Earth's gravity. However, a stone that has a constant force being applied on it need not have an escape velocity. A stone with a thruster with infinite fuel can travel as slow as 1 m/s and still escape Earth.
So now imagine a black hole. Let's name it “Black Hole $001$”. With just enough gravitational pull that it prevents light from escaping.
Photons once generated from the source have an inbuilt energy much like the stone and thus can't escape a black hole's pull. But shouldn't our previous 'stone with a thruster' be able to escape any classical (= non-quantum) black hole as long as it generates a force slightly more than the gravitational force of Black Hole $001$ on the stone?
Is this possible?
 A: As long as it never crosses the event horizon, sure.  Once it crosses the event horizon, it's gone forever.  Gravity isn't really a force.  It's a side effect of the curvature of space.  Once you cross the event horizon, space becomes so curved there is simply no path out anymore.  All worldlines point towards the singularity at the center of the black hole.
A: Here are two thoughts that might help your intuition.


*

*In special relativity, if you have a body subject to a constant force then it has a constant proper acceleration. It accelerates and accelerates. But its velocity still only tends to $c$ (relative to anything else).

*In the case of a black hole, once a body is inside the horizon it is carried to the singularity just as surely as you and I are carried into next week. It is not so much a question of slow and fast, as future and past. For events inside the horizon the whole future lies within the horizon. Once you have crossed it from outside, the horizon lies in your past. So rather than 'once inside the horizon' it might be better to say 'after crossing the horizon'.  
A: I'm +1 on the answer by Ryan_L because I think that represents a very reasonable interpretation of the intent of your question with an answer that is correct within that interpretation.  After some of the comments on the question, though, here's an answer taking a different interpretation of your question.
The premise of the question is that your stone can have "infinite fuel" and also (implicitly) have finite mass. (If it had infinite mass, then by $F=ma$ no finite force would be great enough to accelerate it, not to mention that the center of mass of the Earth-stone system would be at the stone, so you'd be thinking in that case of the stone moving the Earth.  I suppose with "infinite" fuel you could also hypothesize an "infinite" force generated, but I won't go there because by then you've left mainstream physics.) 
Even in the Newtonian case, that's not possible, of course.  But if we somehow accept it, in the Newtonian case, perhaps as an approximation to a fuel with an extreme energy density, then the presence of this fuel does not change the dynamics.  Newton's laws, we assume because we're doing physics in the  Newtonian framework, apply and you can use $F=ma$ to move your stone, maybe, if you're tracking everything, using the rocket equation to account for the change in mass as fuel is spent.
In the case of general relativity, however, we care about the total mass-energy density for determining the spacetime in addition to the dynamics of your stone within the spacetime.  So your stone with "infinite fuel" (whatever that means) is presumably going to curve spacetime itself by quite a bit. Your asserted amount of fuel now either creates a consistency issue or changes the problem drastically from what you originally setout to describe.  The latter possibility is in contrast to the Newtonian case where the assumption was suspect but did not change the equations of motion.  Now the more you try to add fuel to escape, the more you change the spacetime structure, which in turn increases the amount of fuel that you need.  Even as a limiting process, I suspect that diverges or otherwise does "nasty" things that are beyond what you intended to probe with your question.
A: While we can look at Newtonian physics and see that an object with escape velocity greater than $c$ would give rise to many of the properties of what we now know as blacks holes, the  behavior of black holes is more complicated than what is visible from a Newtonian perspective. The fact that objects can't escape from a black hole is much more than just "everything is travelling slower than the escape velocity". 
Even if it weren't from the general relativity effects, special relativity effects such as objects gaining mass-energy as they accelerate, and mass-energy approaching infinity as an object approaches $c$, would prevent an object from escaping. Again, this doesn't take into account the actual general relativity effects, but since you presumably aren't familiar with that level of physics, the following addresses why even without them, your idea of a rocket escaping a black hole doesn't work. 

A stone with a thruster with infinite fuel can travel as slow as 1 m/s and still escape Earth.

Infinite fuel is not possible. So I'll interpret this as "sufficient fuel". How much is sufficient? Well, if it were already travelling at escape velocity, then a certain percentage of its mass-energy would be made up of its kinetic energy component. Call that percentage $p$. For a rocket starting at zero velocity to escape Earth, the percentage of its mass that is fuel must be at least (in reality, due to inefficiencies, much more than) $p$.
As an object approaches the speed of light, the percentage of its mass-energy that is kinetic energy approaches 100%. So if the escape velocity of a body is $c$, then a rocket starting at zero velocity at its surface would have to be made up of nothing but fuel and have a way of converting that fuel to kinetic energy with perfect efficiency. Once the escape velocity exceeds $c$, more than 100% of the rocket would have to fuel.
This shows that this sort of analysis breaks down when you have an object massive enough to be a black hole. To understand exactly how it breaks down, you need to understand general relatively, but at the very least this shows that what you propose doesn't work.
A: A rocket above a black hole burns during one millisecond (time at infinity) one ton (proper mass) of fuel, aiming the exhaust to the black hole. The mass of the black hole increases by 500 kg (as measured at infinity).
Another rocket at lower position than the previous one burns during two milliseconds (time at infinity) one ton (proper mass) of fuel aiming the exhaust to the black hole. The mass of the black hole increases by 250 kg (as measured at infinity).
Those two rockets are supposed to be identical. They burn the same proper mass of fuel during the same proper time.
Observes inside those two rockets feel the same force. So we say that the two rockets produce the same proper force.
If the lower rocket does not ascend but the upper rocket does, we can see, observing from a far away position, various reasons for that:
1) exhaust has less mass
2) fuel burns slower
3) force of gravity is larger
The third one is less important than the two other ones. Gravitational force changes by a small amount when the distance to the center of the black hole changes by a small amount, while time dilation approaches infinity as the distance to the event horizon approaches zero.
