Solution:
Answer: $\frac{1}{2^{4016}}$

Rewrite the equation to get
$$\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}}}=\sqrt{x}+1$$
Squaring both sides yields
$$\sqrt{4 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}=2 \sqrt{x}+1$$
Squaring again yields
$$\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}=4 \sqrt{x}+1$$
One can see that by continuing this process one gets
$$\sqrt{4^{2008} x+3}=2^{2008} \sqrt{x}+1$$
so that $2 \cdot 2^{2008} \sqrt{x}=2$. Hence $x=4^{-2008}$.
It is also easy to check that this is indeed a solution to the original equation.