Abstract: When does an "infinite tower of exponentials" converge? To clarify, for positive real numbers $a,b, \ldots$ let us set $E_a(x) = a^x$, and $E(a,b, \ldots, c) = E_a \circ E_b \circ \cdots \circ E_c (1)$, so $E(a) = a$, $E(a,b) = a^b$, $E(a,b,c) = a^{b^c}$. The question then is: for what sequences $\{a_n\}$ of positive real numbers does the sequence $\{E(a_1, \ldots, a_n)\}$ converge?