To be continued

Let $x$ be the given number. Then

$\large \begin{aligned} x & = 2 + \frac 1{0+\frac 1{1+\frac 1{8+\frac 1x}}} \\ & = 2 + \frac 1{0+\frac 1{1+\frac x{8x+1}}} \end{aligned}$

$\begin{aligned} \ \ \ & = 2 + \frac 1{0+\frac {8x+1}{9x+1}} \\ & = 2 + \frac {9x+1}{8x+1} \\ & = \frac {25x+3}{8x+1} \end{aligned}$

$\begin{aligned} \implies x(8x+1) & = 25x+3 \\ 8x^2 - 24x - 3 & = 0 \\ \implies x & = \frac {24 + \sqrt{24^2-4(8)(-3)}}{2\times 8} & \small \color{#3D99F6} \text{Note that }x > 0 \\ & = \frac {6+\sqrt{42}}4 \end{aligned}$

Therefore, $\dfrac {bc}a = \dfrac {42\times 4}6 = \boxed{28}$ .