How to prove that we are living in a 3+1D world?
This question has changed in such a way that my answer (previously here) didn't seem even related anymore. I therefore came up with something new, gladly inheriting 4 upvotes, but much less confident. In fact, I can plainly state that I'm fully incompetent in these matters.
With that out of the way, another introductory remark. Science doesn't prove things. Dimensions (in this case) are part of mathematical models that aim to describe nature. Whether dimensions "exist" or not is a philosophical question.
Now, let's see...
Anthropic arguments are frowned upon by many, but that is when they are used to explain something. Here one can use anthropic arguments just to reason our way to, not why, but rather, that something is the case. (I guess these arguments aren't really anthropic, but rather anthropic-like.)
My main source, as usual, is Wikipedia. The thing in which we are interested is spacetime (note: a mathematical model). From there we pick up this:
How many dimensions are needed to describe the universe is still an open question. Speculative theories such as string theory predict $10$ or $26$ dimensions (with M-theory predicting $11$ dimensions: $10$ spatial and $1$ temporal), but the existence of more than four dimensions would only appear to make a difference at the subatomic level.
Then the section "Privileged character of 3+1 spacetime" gives us a whole bunch of reasons of the following form:
"if $N\neq3$, then the world (or stuff in it) as we know it wouldn't exist",
where $N$ does not include compactified dimensions invoked by string theory and undetectable to date. (See, e.g., Calabi-Yau manifolds.)
Planets wouldn't have stable orbits. Stars wouldn't have stable orbits. Electromagnetism wouldn't work (at all). Electrons would either fall into the nucleus or disperse. Nerves cannot cross without intersecting. (Some of these arguments apply to $N<3$, $N>3$, or both.)
Hence anthropic and other arguments rule out all cases except $N = 3$ and $T = 1$ [...]—which happens to describe the world about us.
And then there is @Dilaton's comment to the question: "[M]easure how the gravitaional force depends on the distance". That may be expanded upon a little:
"[I]t is the three-dimensionality of space that explains why we see inverse-square force laws in Nature [...]" (Barrow 2002: 204). This is because the law of gravitation (or any other inverse-square law) follows from the concept of flux and the proportional relationship of flux density and the strength of field. If $N = 3$, then $3$-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of radius $r$ has area of $4\pi r ^2$. More generally, in a space of $N$ dimensions, the strength of the gravitational attraction between two bodies separated by a distance of $r$ would be inversely proportional to $r^{N-1}$.
For completeness' sake, I mention also i) the model with large extra dimensions and ii) causal dynamical triangulation, both of which I know even less of. The latter does apparently do something that might appeal to you:
It does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. It shows spacetime to be two-dimensional near the Planck scale, and reveals a fractal structure on slices of constant time, but spacetime becomes $3+1$-d in scales significantly larger than Planck. So, CDT may become the first theory which does not postulate but really explains observed number of spacetime dimensions.
I would bet that the status of this approach (CDT) is controversial.
Now, you stipulated: "without the premise [or] the conception of limited dimensions". I don't know what that really means. I hope that it doesn't mean "without the idea that mathematics can describe nature".
I also hope that I got at least something right. :)
Verify any inverse-square law process, like gas diffusion or classical forces.