Which of the following regular expressions generate(s) no string with two consecutive 1’s?
III is $(0+1)^* (0+ \epsilon)$ which means pick a word from $\Sigma^*$ where $\Sigma = \{0, 1 \}$ and then concatenate it with either $0$ or $\epsilon$.

so III Does generate a string with two consecutive $1$'s. In fact it generates every string which contains $1$'s, $0$'s or is empty.