Weird exponent

We can easily show that $x^{x^{x^{...}}}=\frac{2}{3}$ . Notice that this can be expressed as $x^{\frac{2}{3}}=\frac{2}{3}$ . Raising both sides of the equation by $\frac{3}{2}$ , we get $x=\frac{2\sqrt{2}}{3\sqrt{3}}=\frac{2\sqrt{6}}{9}$ . We now know $a,b,$ and $c$ , so simple multiplication gives us $\boxed{108}$ .

Because $\log_8 4 =\frac{2}{3}$ we have $x^{x\cdots}= \frac{2}{3}$ or $x^{\frac{2}{3}} = \frac{2}{3}$ Solving, we find $x=\frac{2\sqrt{6}}{9}$ so $abc=108$