Is there a maximum accuracy for positions in the universe? I was thinking how, since an object in our universe can move from one position to another, it must have passed through all the positions between those two positions. (I am thinking it moved it a straight line)
This must mean that in actuality there are only so many positions between those two points doesn't it? There must be some maximum accuracy to the concept of positions in space for our universe, because the object has to eventually reach the destination point, no matter how small of a velocity you give it.
Does that suggest that the universe has a maximum accuracy when it comes to positions of objects or is my thinking flawed?
The question I was contemplating that led me to this, is whether an object could pass through every position in a defined 3 dimensional area.
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I was thinking how, since an object in our universe can move from one position to another, it must have passed through all the positions between those two positions. (I am thinking it moved it a straight line)
This must mean that in actuality there are only so many positions between those two points doesn't it? There must be some maximum accuracy to the concept of positions in space for our universe, because the object has to eventually reach the destination point, no matter how small of a velocity you give it.

Your idea is ambiguous about what the word "finite" means in this context. If I have a pile of lego bricks, there is a finite number of bricks in the pile. It is not necessarily the case that space must be finite in this sense. All that would matter is whether there is a way of attaching numbers to regions of space that gives finite answers, such a way of attaching numbers is called a measure. The mistake is to think that the word "finite" as you would apply it to a pile of lego bricks has to work when applied to other systems, but this is not necessarily the case.
In addition, if there is a limit to how localised a system can be, that does not in and of itself imply that it doesn't pass through all of a continuous set of positions. If you imagine a line 5cm long and a block 1cm in length, it could move along the line passing through all of the positions even if they are continuous.
Actual physics does say something like there are a finite number of positions. Most current theories of quantum gravity agree that there is a finite number of states that can occur in any finite volume. This is called the Bekenstein bound. And there are bounds on the number of states that various classes of systems can take up:
http://journals.aps.org/prd/abstract/10.1103/PhysRevD.30.1669.
A: The short answer is that we don't know the answer to this question. In physics terms you are asking whether spacetime is quantised or not.
Some theories exist that do quantise spacetime, Loop Quantum Gravity is an important one. Some don't. The problem with all of them is that the scale at which we're likely to find out is so small that no current or practically buildable experiments can answer this directly. It may be possible to come up with experiments that indirectly point one way or another but I'm not aware of them.
On a somewhat related note, what is well established is that, the more accurately you measure the position, the less accurately you can measure the velocity (technically the momentum). This is a consequence of the Heisenberg Uncertainty Principle and is a fundamental result in quantum theory i.e. it's very well established. This doesn't actually prevent you from getting more and more accurate positional measurements but it does mean that you'll have less and less idea where the particle is headed if you do.
One final point. Beware intuition in this field. It's pretty straightforward to intuit answers to these sorts of questions. History has shown, however, that our intuition tends to be completely wrong at very small scales. This is definitely one area where applying logic and reason is trumped by careful experimentation every time. 
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*"Does that suggest that the universe has a maximum accuracy when it comes to positions of objects or is my thinking flawed?" Yes, there is a maximum accuracy, beyond which we cannot know positions, or velocities, or any other physical quantity. Look up the Heisenberg uncertainty principle. 

*"The question I was contemplating that led me to this, is whether an object could pass through every position in a defined 3 dimensional area." Look up the path integral formulation of Quantum mechanics by Feynman. 
Please ask this question at the physics stack exchange for more details. 
A: There are objects in the universe yet to be discovered as shown by constant new discoveries of new types of planets under conditions that were previously considered impossible. (Diamond Planet, Fire Ice Planet)  I believe this works the same way with objects in motion as well.
If the Big Bang theory is correct, and our current interpretation of how physics works here, applies to everything in the universe (it doesn't : eg,  Quantum Entanglement), then by that, you can (with a really big computer), calculate the finite directions, trajectories, future and past collisions, planet/star/galaxy/etc formations, and so forth.
Further, given that the universe is expanding, this means the distance traveled is also getting longer at a rate proportional to expansion. In theory, if the universe is expanding a rate faster than the trajectory of an object, this would mean the object will potentially never complete it's route (eg, a straight line) unless the expansion stopped or the object increased velocity passing the expansion rate, or the expansion slowed down to less than the current velocity of the object.
By this definition, the maximum accuracy would not be able to be calculated as it would be dependant on their being a finite amount of area in the universe, and the distance to travel (a straight line), is fixed.
