Based on your second paragraph, I assume you ask if the following is true or not:
Yes, it holds. The indices doesn't matter. Call them $x_1=a,x_2=b,x_3=c$ and you won't have a standard form. To be specific, you'll have $n!$ different factorizations if you have $n$ RVs.
The intuition is you can always group RVs as if they're single, e.g. $x_1=a, (x_2,x_3)=b$. When you apply your standard chain rule formula you can end up in any possible factorization.
Note: $p(x),f(x)$ are all abuse of notations. A better notation is to use something like $p_X(x)$, to indicate both the RV and the specific value.