If $x_0 = 1/2$ and $x_{n+1} = \sqrt{\dfrac{1+x_n}{2}}$, it appears that $$ \prod_{n=0}^\infty x_n = \dfrac{3 \sqrt{3}}{4 \pi}$$ (verified numerically to $300$ decimal places, but I don't have a proof)
Robert Israel
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