No need to use calculator

By the binomial theorem , for $x \gt 0$ and positive integer $n$ we have that

$(1 + x)^{n} = 1 + nx + \dfrac{n(n - 1)}{2}x^{2} + O(x^{3}) \gt 1 + nx$ .

Thus $1.01^{1000000} = (1 + 0.01)^{1000000} \gt 1 + 1000000 \times 0.01 = 10001 \gt 10000$ ,

and so $\boxed{A \gt B}$ .