Observe an alternating convergent recurrence

Use sequence mode with $x_1=-2$ and $x_{n+1}=-\exp(x_n)$. The source calculator table is reproduced to three decimal places:

The terms alternate on either side of a limiting value, so the full sequence is neither increasing nor decreasing. The calculator values approach $-0.567$ (3 significant figures). A fixed point satisfies $\ell=-e^\ell$; numerical solution gives $\ell\approx-0.56714329$. The table is numerical evidence of this behaviour, not by itself a proof of convergence.