jmc

Let $b_n=\frac{1}{a_n}.$ Then $a_n=\frac{1}{b_n},$ so $$\frac{1}{b_n}=\frac{\frac{1}{b_{n-1}}}{1+\frac{1}{b_{n-1}}}=\frac{1}{b_{n-1}+1},$$so $b_n=b_{n-1}+1.$ Since $b_0=\frac{1}{6},$ it follows that $b_n=n+\frac{1}{6}$ for all $n.$ Hence, $$a_n=\frac{1}{n+\frac{1}{6}}=\frac{6}{6n+1}.$$for all $n.$ It follows that $a_{100}=\boxed{\frac{6}{601}}.$