Does this nested exponential summation converge?

[neom]

$$\sum^k_{n=1}e^{-n\sum^k_{n=2}...e^{-n\sum^k_{n=k-1}e^{-n}}}$$

Can anyone help me find out if this converges and if so how to calculate the sum?
I don't have an idea on how to even start.

This is not homework

 

[neom]

Anyone?