smc

We compute: $$\begin{array}{rl}1+\frac{1}{1+\frac{1}{1+1}}& =1+\frac{1}{1+\frac{1}{1+1}}\\ & =1+\frac{1}{1+\frac{1}{2}}\\ & =1+\frac{1}{\frac{3}{2}}\\ & =1+\frac{2}{3}\\ & =\frac{5}{3}\end{array}$$ This is choice $\boxed{\frac{5}{3}}$. Interesting sidenote: The continued fraction $1+\frac{1}{1+\frac{1}{1+\mathrm{1....}}}$ is equal to the golden ratio, or $\frac{1+\sqrt{5}}{2}$.