Answer
The domain of $ r(t) $ is $$ D=\{t\in R; t\neq 0, -1\}.$$
Work Step by Step
Given $$ r(t)=e^ti+\frac{1}{t}j+(t+1)^{-3}k $$
We know that $1/t $ is not defined for $ t=0$ and $(t+1)^{-3}$ is not defined for $t=-1$ (because we can not divide by $0$). Thus, the domain of $ r(t) $ is $$ D=\{t\in R; t\neq 0, -1\}.$$
