The limit of $\lim\limits_{n\to\infty} \exp(-1+\exp(-2+\exp(-3+\cdots\exp(-n) \cdots)))$.
The limit exists because the sequence is monotonic and bounded, although I don't know yet what the limit is. Let's denote this sequence $E_n$.
The sequence is monotonic because $-n+\exp(-(n+1))>-n$, therefore $\exp(-n)+\exp(-(n+1)))>\exp(-n)$, therefore $\exp(-(n-1)+\exp(-n)+\exp(-(n+1))))>\exp(-(n-1)+\exp(-n))$, etc, until you get $E_{n+1}>E_n$.
The sequence is bounded because $-n+\exp(\text{negative number})<0$, therefore $E_n<\exp(0)=1$.
By monotone convergence theorem $E_n$ converges.