Prelude: Problems
What numbers have names?
When a number doesn't have a name, how often do we point to it by saying how to calculate it?
In the Roman numeral system, what numbers have symbols, and which numbers are indicated by how to calculate the number?
In the stick system, what numbers have symbols?
Of Names and Numbers
Children start out learning the whole numbers. So, for example, they know there is a number 3 and a number 4, and that 3 comes after 4. The modern idea, however, is that there is a lot of numbers in between 3 and 4, whether or not we assign names to them. Anyway, this important concept raised the question, what numbers have names? Clearly the numbers 0 to 10 have their own names, and pi has its own name, but what other numbers have their own names?
I decided that "ninetyeight" did not count as a name. Obviously, it was a derivative of nine and eight. The strictest criterion for a name is that it is an arbitrary association between a sound and the value it represents. Eight is arbitrary, and nine is arbitrary, and other languages have different sounds for these values. "Ninetyeight" is not arbitrary.
At first, this was just a fun game. What numbers were worthy enough to have their own names? It was fun to assume I had found all the numbers with names and then to find new ones.
It was also fun to think about the origin of numbers. I was calling eleven a name, but it is derived, esentially meaning a pile of 10 and 1 left over. (Same for twelve.) So I once was disqualifying it as a name. But I decided that the origin had been lost, so now I count it. (This isn't exactly a problem with right answers.)
When There is No Name: Process
I eventually realized that if a number didn't have it's own name, we called it by how it could be calculated. So, "ninetyeight" means 9 times 10 plus 8.
This new perspective made more sense of fractions. Consider the simple fraction 2/5. This has two (or three) very different meanings. First, it stands for a process  2 divided by 5. But it also stands for a real number. The reconciliation is that we point to real numbers by the process of calculating them. Now it seems as plain as the nose on my face that 2/3 is both a process and a number, and we use the process of calculating the number to indicate the number. Same thing for 3 and the square root of 2.
Other Systems
To use this analysis on the Roman Numeral System, they had essentially arbitray symbols for a few numbers. Those symbols were I,V,X,L,C,D,M. The value of ninety eight did not get it's own symbol. Instead, it was represented by how it could be calculated (LXXXVIII). (And which numbers in our modern system have their own symbol?)
Now let's do the stick system. Three is represented as ; nine is represented as ; and ninetyeight is, well, a whole lot of sticks, you will just have to imagine. In a sense, this representation scheme has just one arbitrary symbol, the  which stands for one. All the other numbers are represented by how they could be calculated. For example three is 1+1+1. And though this is a primitive system, it is also used in the modernday axiomatization of numbers.
My Answer
Adopting the strictest criterion I could, I have 23 number names:
010
eleven and twelve (also known as a dozen)
hundred, thousand, million
score, gross
half
pi, e
googol (10 to the hundreth power), googolplex
Close but derivatives
1319
20,30, etc.
billion, trillion, etc.
quarter, third
I don't count phrases, such as Plank's constant, Avogadro's number, or baker's dozen.
I thought the term googolplex was not commonlyenough known to count as a name. But the professor used it in the movie Back to the Future.
Is infinite a number? Or is it just a concept? Alephnull, alephone, and alephtwo are not arbitrary names but neither are then methods of calculation. Countably infinite and uncountably infinite I would classify as phrases.
i is too tough for me. Obviously, it is an arbitrary name (or at least symbol). It stands for the square root of 1, which is the process of calculating i. But it is not clear to me that there really is a number that this process leads to.
