How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?

For any $n\ge4$, we have $\sqrt{2n} \le n-1$. Therefore \begin{align*} a_n &\le \sqrt{1+\sqrt{2+\sqrt{\ldots+\sqrt{(n-2)+\sqrt{(n-1) + \sqrt{2n}}}}}}\\ &\le \sqrt{1+\sqrt{2+\sqrt{\ldots+\sqrt{(n-2)+\sqrt{2(n-1)}}}}}\\ &\le\ldots\\ &\le \sqrt{1+\sqrt{2+\sqrt{3+\sqrt{2(4)}}}}. \end{align*} Hence $\{a_n\}$ is a monotonic increasing sequence that is bounded above.

The first number $1$ is a nuisance, so at first we disregard it.

We proceed by induction, and deal with finite nested radicals that start with $\sqrt{k+\sqrt{(k+1)+\cdots}}$, where $k\ge 2$.

We will show that such a radical is $\lt 2k$, by induction on depth. The result is certainly true for all nested radicals of depth $1$, since $\sqrt{q}\lt 2q$.

For the induction step, a nested radical of depth $n$ that starts with $k$ is $\sqrt{k+R}$, where $R$ is a nested radical of depth $n-1$ that starts with $k+1$. By the induction assumption, we have $R\lt 2k+2$. But then $\sqrt{k+R}\lt \sqrt{3k+2}\lt 2k$ if $k \ge 2$.

So (finite) nested radicals of any depth that start with $2$ are $\lt 4$. The sequence of nested radicals is clearly increasing, so it converges. It follows that the nested radical of the post is $\le \sqrt{1+4}$.

An easy sloppy way to see it: You know that $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}=\phi$. Now multiply by $2$ to get $2\phi=2\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}=\sqrt{4+4\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}=\sqrt{4+\sqrt{16+\sqrt{256+\sqrt{256^2+\dots}}}}>\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\dots}}}}$. Of course this should be done more rigorous.