Are there some natural contexts in which a double exponential occurs, $x$ to the ($y$ to the $z$): $$ x ^ {(y ^ z)} \;, \textrm{or} \;, a ^ {(b ^ c)} \; \textrm{?} $$ Of course one can contrive many problems that ask computational questions concerning towers of exponents, possibly challenging problems. And in my own area of research, $2^{n^2}$ is not uncommon; EXP is exponential-time $O( 2^ {n^k} )$. Generalized chess falls in this class.
I am more seeking some context in which such a (limited) tower occurs naturally and would be understood and appreciated by high-school or early-college students.