Let's first note that if we are given an expression of the form $1 + \frac{1}{n},$ we can rewrite this as $$\frac{n}{n} + \frac{1}{n} = \frac{n+1}{n}.$$ With that in mind, we can rewrite the expression given to us in the problem, as shown below: $$\begin{align*} &\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\\ &\quad{}\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\\ &\quad{}\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)\\ &=\dfrac{2}{1} \cdot \dfrac{3}{2} \cdot \dfrac{4}{3} \cdot\dfrac{5}{4} \cdot\dfrac{6}{5} \cdot\dfrac{7}{6}\\ &=7 \end{align*}$$ Thus, the correct answer is D.