Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.3 |

Graduate School of Mathematics

Nagoya University

Furo-cho, Chikusa-ku

Nagoya 464-8602

Japan

Kota Saito

Faculty of Pure and Applied Sciences

University of Tsukuba

1-1-1 Tenodai, Tsukuba

Ibaraki 305-8571

Japan

Wataru Takeda

Department of Applied Mathematics

Tokyo University of Science

1-3 Kagurazaka, Shinjuku-ku

Tokyo 162-8601

Japan

**Abstract:**

We say that the limit of a sequence of functions
is the iterated exponential function, denoted by .
By a result of Barrow, this limit is convergent for every
.
In this paper, we prove that, for each fixed integer ,
the limit is transcendental
for all but finitely many algebraic numbers
with
.
Furthermore, let be the cardinality of exceptional points . We prove that the ratio
approaches as
, where
denotes Euler's totient function.

Received December 24 2022; revised versions received March 8 2023; March 11 2023; March 12 2023.
Published in *Journal of Integer Sequences*,
March 13 2023.

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