Some analysis and computation in wolfram alpha show that sequence $a_n=-\frac1e+e^{(\frac1e)^{...^{\frac{1}{e^n}}}}$ could be close to integer exactly $2$ as shown here for $n$ even and for $n$ is odd it gives approximately 1.135.. , Now in that case I'm mixed to predict its limit , then I should say the limit is not exact , But probably i will have a luck to get limit close to $2$, then my question here is :
Question: What is the exact limit of $a_n=-\frac1e+e^{(\frac1e)^{...^{\frac{1}{e^n}}}}$ for $n \to \infty$ ?