Have two experiments ever been done such that the set up is identically the same, (when I say identically I mean down to the atomic level). Of course the experiment would have to be very simple. If it has been done do we see differences in the results (as would be expected with the probabilistic nature of quantum mechanics)? If not, what are the limitations preventing us from doing such an experiment?
Sure! In fact, this is one of the great strengths of atomic physics--if you and I are doing two experiments on rubidium atoms then (assuming we use the same isotopes) then our atoms will be exactly the same. Your rubidium atom and my rubidium atom are simply identical. So plenty of atomic physics is very repeatable, and such.
As for the probabilistic aspect, this is not really the right way to look at "an experiment." Suppose we have have an experiment that's the quantum equivalent of a coin-flip. Maybe we excite an atom in such a way that it can emit either $A$ light or $B$ light with a 50% probability. (If you're more familiar with quantum mechanics, a better example is a Stern-Gerlach experiment.) Ok, so we load the atom into a trap, hit it with a laser, and wait to see whether the $A$ or the $B$ detector clicks. That's the experiment.
Thing is, this experiment kind of sucks if I only do it once. So I do it again, and I get something else--probabilistically, as you say. But really, I can probably do this thousands of times a second, so I do it for a long time and I count how many clicks I get and I get 50% on each detector. For the experiment to actually tell me something, I have to do it over and over again. In some sense, then, I am "repeating" my experiment many times with the exact same setup (unless you really don't trust that the laser is stable over the course of microseconds). But if you told someone you'd done "an experiment" and only measured one atom-flash, they would probably raise an eyebrow. In addition, I've left out the fact that very rarely does anyone do an experiment on a single atom. In reality you probably excite many atoms at once, which is also kind of "repeating the experiment" (except that you might have to take into account interactions/superradiance).
If my friend down the hall wants to repeat my experiment, she can do the same set up, and excite the same atom, but chances are she won't just take one single data point--she too will take lots and try to show that it works out to be precisely 50% in each detector. That sort of averaged statistical data is far more valuable to us as scientists, and it doesn't change from experiment to experiment.
The simplest answer to your question is probably the single-slit experiment. If you shoot two electrons consecutively in the same direction towards a plate pierced with a slit, then two dots will appear at different positions on the screen placed behind the plate. The difference in the positions is almost entirely due to the quantum mechanical uncertainty. This has really been done and, in fact, the more well-known but slightly more complex double-slit experiment is one of the most famous experiments for testing quantum mechanics.
If care is taken and not too much time passes between the events, then we can say that the electrons indeed experience exactly the same experimental setup. I see from the comments that you also require the atoms or, presumably in this case, the electrons to be the same for both measurements. I don't think this is possible due to the principle of indistinguishability. I mean, it would be impossible to pick up exactly the same electron for repeating the experiment.