A simple way to approximate $e$

\begin{align} \frac{22}{7} &= 3.14285714285714…&{} &\approx \pi \\ e^\pi - \pi &= 19.9990999791894…&{} &\approx 20 \end{align}
\begin{align} &\Rightarrow & e^{\frac{22}{7}} - \frac{22}{7} &\approx 20 \\ &\Rightarrow & e^{\frac{22}{7}} &\approx 20 + \frac{22}{7} \\ &\Rightarrow & e &\approx \sqrt[\frac{22}{7}]{20 + \frac{22}{7}} \\ &&&= (20 + \frac{22}{7})^{\frac{7}{22}} \\ &&&= 2.7172692474731… \end{align} Close enough.