Rewrite the relation as $$\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\cdots+\sqrt{4^{2008}x+3}}}}}=\sqrt{x}+1$$ Square both sides and simplify $${\sqrt{4x+\sqrt{16x+\sqrt{\cdots+\sqrt{4^{2008}x+3}}}}}=2\sqrt{x}+1$$ Square both sides and simplify $${{\sqrt{16x+\sqrt{\cdots+\sqrt{4^{2008}x+3}}}}}=4\sqrt{x}+1$$ $$\cdots$$ $$\sqrt{4^{2008}x + 3} = 2^{2008}\sqrt{x}+1\implies 2\cdot 2^{2008}\sqrt{x}=2\implies x=4^{-2008}=\boxed{\frac{1}{2^{4016}}}$$