$$\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}$$
My approach
1st approach: Let $\sqrt{1+\sqrt{2+a}} = x$, where $a$ is a result of the lower portion of the continued nested radical and $1+\sqrt{2+a}$ is a perfect square, and $x$ is a number that doesn't contain a radical.
Solving for $a$ gives 7, and $x$ equals to 2.
2nd approach: Using a software, $\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}} \approx 1.75793...$
What is the best approach for the nested radicals?