Fractions

$\displaystyle\frac{3}{8}=\frac{1}{\frac{8}{3}}=\frac{1}{2+\frac{2}{3}}=\frac{1}{2+\frac{1}{\frac{3}{2}}}=\frac{1}{2+\frac{1}{1+\frac{1}{2}}}$ .

Thus, the answer is $2\cdot1\cdot2=\boxed{4}$ .

From the other side :

1/(a+(1/(b+(1/c))))=1/(a+(c/(bc+1)))=(bc+1)/(abc+a+c)=3/8

So bc=2 , thus c=1 or 2

And 3a+c=8. The only possibility is c=2, a=2, b=1...