But That's Huge!

Relevant wiki: Properties of difference of two squares

Observe that multiplying by $(2-1)$ will make no difference to the complete product, but observe:

$(2-1)(2+1)(2^2 +1) = (2^2-1)(2^2+1) = (2^4-1)$

This triggers a chain reaction.

Continuing in the same way, we reach:

$(2^{1024} -1)(2^{1024}+1) = 2^{2048}-1$

$n = 2048$

Identity used: $(a+b)(a-b) = a^2 - b^2$

Relevant wiki: Properties of difference of two squares

Notice that $2+1 = 3 = 4-1 = 2^2 -1$

Substitute this in:

$(2+1)(2^2+1)(2^4+1)(2^8+1)\ldots(2^{1024}+1)\\ =\color{#3D99F6}{(2^2-1)(2^2+1)}(2^4+1)(2^8+1)\ldots(2^{1024}+1)\\ =\color{#EC7300}{(2^4-1)(2^4+1)}(2^8+1)\ldots(2^{1024}+1)\\ =\color{#D61F06}{(2^8-1)(2^8+1)}\ldots(2^{1024}+1)\\ =(2^{16}-1)\ldots(2^{1024}+1)\\ =(2^{1024})^2 - 1\\ =2^{2048}-1$

Therefore, $n=\boxed{2048}$