We can greatly simplify this problem by dividing the expression by $\sqrt{5}$. This will leave us with $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}$$. The expression $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}$$ is well known to have a value of $\phi$ (see %%HREF%%3882%%), i.e. $\frac{1+\sqrt{5}}{2}$. Multiplying $\phi$ by $\sqrt{5}$ will give us the orginal expression, equal to the answer of $x=\boxed{\frac{5+\sqrt{5}}{2}}.$