Example 9 The solution to the equation $\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+x}}}}=x$ is $\qquad$ .
Analysis: It is difficult to proceed with removing the square root to solve. Instead, compare the size of $x$ and $\sqrt{3+x}$. If $x\sqrt{3+x}$ also leads to a contradiction with the given information.
Therefore, we have $x=\sqrt{3+x}$, solving this yields $x=\frac{1+\sqrt{13}}{2}$.