The following is a partial English translation of the 1901 paper "Über die Funktion y = x(x(x(...))) für ganzzahliges Argument (Abundanzen)" by Hans Maurer, the first ever paper describing tetration. The paper also features a very early reference to the p-adic numbers, only 2 years after their discovery by Kurt Hensel in 1899. The original source (in German) can be found here.

About the function y = x(x(x(...))) for integer arguments (Abundances)

The function mentioned in the title, which one reaches through a fourth-order operation, may be denoted by the symbol y = nx (read: x left up to n), where the number n indicates how often one must exponentiate x by itself. Thus, for example, 3x would be x(xx). The function y = nx is called the nth abundance of x or the abundance of nth degree of x; x is the base of abundance, n is the exponent of abundance or degree of abundance. The term "exponent" alone in this work always denotes a power exponent. According to the definition, every abundance arises by raising the base to a power, the exponent of which is the abundance one degree lower. So, in general: nx = x(n-1x). If this relationship is also valid for values of n<2, it serves to define the expressions:

0x = 1; -1x = 0; -2x = -∞

1x = x can also be inferred from the same relationship (n = 2), if one does not want to consider this as already part of the original definition. If one forms the equation of relationship for n = ∞, it results in:
∞x = x(∞x) or (∞x)√∞x = x

The name "abundance" is justified when one considers the overwhelming rate with which the function output increases as the base or the exponent of abundance increases. It is: 22 = 22 = 4; 33 = 327 = 7,625,597,484,987.
The characteristic of log(43) would thus already be > 3.6 billion [This paper uses the long scale, where billion = 10^12, not 10^9], and the number 43 itself, calculated at 4 digits per centimeter, if written out in a row, would circle the Earth's equator 227 times. About 44, Krönig notes in his book "The Existence of God and the Happiness of Men" that a sphere, whose radius is a quintillion (= 10^30) light-years long, would not be able to contain the printer's ink required to print the number 44 in the smallest existing letters. While it soon becomes virtually impossible to specify even the first digit of such an abundance, it requires relatively little calculation to determine the last digits, the ones, hundreds, etc., of all abundances of an integer. It even turns out that every abundance of the form ∞x, the value of which is naturally ∞, has a certain sequence of digits, and that their ones, tens, etc., can easily be specified. This is based on the fact that the powers of whole numbers with increasing exponents repeat their last digits from certain values of the same.



Bonus: A translation of a section of "Das Dasein Gottes Und Das Gluck Der Menschen", written by August Krönig in 1874 (The section is from 1872). The original source (in German) can be found here.

The Existence of God and the Happiness of Men; Chapter 131: About the Infinite

Could my inability to comprehend the infinite perhaps stem from a certain sluggishness of my imagination, which would not be capable of rising above the clumsiness of our usual numerical designation through the decimal system? I believe I can answer this question with no. One could perhaps designate the number 11 by U(1), the number 22 by U(2), the number 33 by U(3), and so on. Then U(10) would already be equal to ten billion, U(100) would be a number written with a one followed by ten thousand zeroes [This is incorrect. Krönig guessed that 100^100 = 10^(100*100) = 10^10,000, but the correct value is 100^100 = 10^(100*log10(100)) = 10^200]. Would the above-mentioned number a, the number of particles in a gram of iron, be very large? As such, I cannot conceive of there not existing a U which would be larger than a. If one would not consider the above-described numbers U large enough, one could denote by V numbers of the kind that V(2) is equal to 2^2 or 2 raised to the power of 2, V(3) equals 3^3^3, V(4) equals 4^4^4^4*, and so forth. Then, V(2) equals 4, but V(3) is already greater than 7 trillion. An approximate conception of V(4) can be made in the following manner: Imagine a straight line of such length that a beam of light, which covers 42,000 miles in a second, would need a quintillion - written as one and thirty zeros - years to traverse it. Furthermore, consider a sphere described with this line as its radius and the volume of the sphere filled with printer's ink. This latter quantity would not suffice to print the number V(4) calculated according to the decimal system with the smallest existing letters legibly.

*Since all calculation consists of the successive combination of two numbers into one, it is easy to see that an expression composed of n numbers must contain n−2 parentheses, which indicate the order of the successive combinations. By tacit agreement, rules have been established for the commonly occurring number combinations or expressions, specifying which parentheses must be assumed in expressions that are written without parentheses. For number combinations of the type described above, however, this has not yet been done. Since it is important to me to make V quite large, I now propose that they should be calculated in reverse, starting from the end, or to use common mathematical notation. [This footnote is regarding the fact that exponentiation is not associative, and suggests working top-down (right-associative) for the V function, which is also the accepted method today.]

This would then be V(4). If the task were put to me to describe V(5) in a similar manner, I truly wouldn't know how to begin.

All the following V's are even more unfathomable. And yet, if the above-described a, the number of particles in a gram of iron, were larger than V(1000), but smaller than V(1001), then it would undoubtedly be enormously large, but not infinite. An infinite a, as I have already said, and as I will further justify below, I cannot comprehend. However, it is very possible that precisely on this question, an objection will be raised against me that the number in question, the number of particles in a gram of iron, is infinitely large, and must be so due to the infinite divisibility of matter.