<s>[INST] How do I solve this infinitely nested radical? $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$
Apparently, the answer is 3.
 [/INST] Hint:  Observe $$(x+1)^2 - 1 = x(x+2),$$ or $$x+1 = \sqrt{1 + x(x+2)}.$$  Now recursively substitute:  $$\begin{align*} x+1 &= \sqrt{1 + x\sqrt{1 + (x+1)(x+3)}} , \\ &= \sqrt{1 + x \sqrt{1 + (x+1)\sqrt{1 + (x+2)(x+4)}}}, \\ &= \sqrt{1 + x \sqrt{1 + (x+1)\sqrt{1 + (x+2)\sqrt{1 + (x+3)(x+5)}}}}, \ldots \end{align*}$$  Of course, you need to do something much more rigorous to complete the proof.  I leave that to you.
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