Finding your own roots

In South Korea, the numbers "2" and " $e$ " are pronounced the same, so it's impossible to pronounce these numbers distinctly: $2^2,\ 2^e,\ e^2,\ e^e.$ It is obvious that the first number $2^2$ is the smallest, while the last number $e^e$ is the largest. But, are they all arranged in ascending order?

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Notation: $e = \frac1{0!} +\frac1{1!} + \frac1{2!} + \cdots \approx 2.718$ .