One is arguably the most important of all numbers. It is defined either as the number x such that any number a multiplied by x is still a, or alternately as S(0) with the fundamental successor function (see this page for details). This means that 1 is the multiplicative identity, and like 0 it's an inherently important number. One therefore has countless mathematical properties, way too many to list and mostly for trivial reasons. For example, it often leads to degenerate cases like 0 does: 1*x = x, 1^x = 1, 1^^x = 1, 1^^^x = 1, etc, with as many up-arrows as we please. Conway chain arrows and Bowers' arrays starting with 1 also solve to 1, as do 1's put into polygons in Steinhaus-Moser notation, Hollom hyperfactorial expressions beginning with 1's, and so on - this kind of thing happens with ones in almost all googological functions. Some other examples of degenerate cases leading to or involving 1 are: x^0, x^^0, x^^^0, etc. = 1, x^1, x^^1, x^^^1, etc = x, Bowers' arrays with 1 as the second entry evaluate to the first entry, and all fast-growing hierarchy expressions of the form fα(1) evaluate to 2.
Logarithms of base 1 are undefined because 1 to the power of any number will always be 1 - therefore 1 is also the first square number, cube number, perfect tesseract, etc - it's the only positive integer expressible as a perfect power in infinitely many ways, and 0 also has this property. A lot of the trivial properties of 1 come from 1 being used as the first member of many sequences - for example, the Fibonacci sequence starts with 1, as do the prime recursion sequence, the triangular numbers, and so on.
One is the only number out of all numbers that represents a single quantity instead of a multiple quantity. While all other numbers are plural (-6 votes, 1.001 grams, 35 people, -1 feet), one is singular (1 vote, 1 gram, 1 person, 1 foot).
If there is any counting number that is never ever EVER large, it’s one. We tend to perceive an object more as being a single object than as being only one object. People probably thought of the concept of twoness and then came to recognize oneness. Before that people didn’t think of one, just the immediately recognizable two, three, and four.
One is an image that rests clearly in the mind as a single object, with some numbers still being just as clearly put in the mind. One is also an example of what can be perceived with our basic number although it's really is a trivial case. For example, if you look at one giraffe, you don't think "one giraffe", you think "a giraffe".
Sbiis Saibian proposes that one should be the boundary between small and large numbers - this is convenient because the small and large numbers are mirrored accross each other with the reciprocal function, and there is no arbitrariness like if we made, say, 100, the smallest large number. To me, that really solves the problem of what is the smallest large number. Therefore the number one can be thought of as a natural place to start any journey through very large numbers.
1 is vital in forming large numbers though because of its use as the smallest possible argument in many googological functions. For example, the rules of Bowers' arrays work quite interestingly when dealing with 1's that do not end an array, and Conway's chain arrows have special rules for when the last or second last argument in the chain is 1.
Many words in English are based on roots for one. The most common prefixes for 1 are uni- (unit, unicycle, unibrow, unify), and mono- (monocle, monotheism). The root uni- also has expanded its meaning to also mean united or together, in words like universe or unisex.
Side note: One can be called googolplei using (my version of) the googolple- prefix.
Now that we’re done with discussing one, we shall slowly go through the numbers between one and two.