And by nice looking law I mean with no constants. I mean, what would we need to set, so all laws would without those nasty constants in front of them? (I mean all of them, also $\pi$!) What would it mean?.
What I am trying to say with this question is: Is there a universal language, do particles and gravity live in the same space? If gravity and quantum world should go together, there should be one basis, where you don't need to use any constants.
Ok, here's why rescaling units won't get rid of $\pi$:
The circumference of a circle is $C = 2\pi r$. Note that $C$ and $r$ are measured in the same unit, both are units of length. Now rescale lengths as $l' = c l$ with an arbitrary constant $c$ (might be $\pi$, but doesn't matter). Plugging in yields
$$C' = cC = c 2 \pi r = 2 \pi c r = 2 \pi r'$$
So you haven't changed the circumference law at all. Thus, you don't get rid of $\pi$.
Planck units is almost (except coefficients like $\pi$) exactly the unit system you want, and it is frequently used in quantum field theory.
According to its definition:
$c = G = \hbar = k_\text{B} = 1 \ $
When you're doing Fermi estimation, it's convenient to use a "system of units" where $$ \sqrt{10} = \pi = 3 = 1 = -1 = c $$ though this is probably more tongue-in-cheek than you were hoping for. For some reason I think of these as "pauper's units," though I'm not sure whether that phrase is my own invention or not.
You might be interested in this other question.
