For positive values of $a$ , the power tower sequence $$a,\, a^a,\, a^{a^a},\, a^{a^{a^a}},\, \ldots$$ is convergent if and only if $$\frac{1}{e^e} \leqq a \leqq e^{\frac{1}{e}},$$ approximately $$0.065989\leqq a \leqq 1.444667.$$ The limit of the sequence is the least real root of the equation $$a^x = x.$$ The proof is found in [1].

Bibliography

1

E. LINDELÖF: Differentiali- ja integralilasku ja sen sovellutukset III. Toinen osa. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).