ABC too easy!

:)

Multiply $bc$ in the equation $a+\frac1b=1$

Giving $abc+c=bc$

Note that $b+\frac1c=1\Rightarrow b=1-\frac1c$

Substituting the value of $b$ gives

$abc+c=c*(1-\frac1c) \Rightarrow abc+c=c-1 \Rightarrow abc=\boxed{-1}$