Assume that X, Y are two bounded random variables. If for any integers $m, n \geq 0$, $E[X^m Y^n] = E[X^m]E[Y^n]$, then X and Y are independent.
I've worked out that, for sure, if $E[f(X)g(Y)] = E[f(X)]E[g(Y)]$ for any functions $f(X)$, $g(Y)$, then they must be independent (which can be proven by assuming $P(X = x$ and $Y = y) \neq P(X = x)P(Y = y)$, taking f, g to be characteristic functions, and reaching a contradiction). However, I can't think of any way to weaken the condition to just whole-number powers, either directly or indirectly (through the "any function" case). I suppose I could try using a Taylor series expansion, but that seems way off-base for this problem, and like it wouldn't even cover most functions, since that really only applies to differentiable ones.
Just notice that you can approximate the characteristic function of (of points) by polynomials. Also notice that if $f,g$ are polynomials, then the above condition gives $E(f(X)g(Y))=E(f(X))E(g(Y))$