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Overview
The following chart is an organization of fractions of 1/n, increasing n by 1 each time until n = 18 decimally. Note that I've only chosen even bases, since odd bases cannot be divisible by 2 without recurrence (ie. 0.2222). This is arguably the most important property in finding our base.
Furthermore, to represent our higher values, I've also chosen to use lowercase letters instead of uppercase, because:
 8 & B
 D & 0
 G & 6 (although this one isn't so bad in most fonts)
 I and 1
...they all look similar.
Another problem arises when you reach letter L. We can't use lowercase L because it looks like 1 and the letter 'I' as in "igloo".
At base 18, we use the letter H, but not I; it becomes a problem beyond base 18.
It is for these reasons that I now only show the bases from 6  18:
Furthermore, to represent our higher values, I've also chosen to use lowercase letters instead of uppercase, because:
 8 & B
 D & 0
 G & 6 (although this one isn't so bad in most fonts)
 I and 1
...they all look similar.
Another problem arises when you reach letter L. We can't use lowercase L because it looks like 1 and the letter 'I' as in "igloo".
At base 18, we use the letter H, but not I; it becomes a problem beyond base 18.
It is for these reasons that I now only show the bases from 6  18:
Senary

Octal

Decimal

Dozenal

1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) 
.3
.2 .13 .1 .1 .05 .043 .04 .0 3 .0313452421 .03 .024340531215 .0 23 .0 2 .0213 .020412245351433... .02 
.4
.25 .2 .1463 .125 25 .1 .1 .07 .06314 .0564272135 .052 .0473 .0 4 .0421 .04 .03607417 .0 34 
.5
.3 .25 .2 .16 .142857 .125 .1 .1 .09 .083 .076923 .07 142857 .0 6 .0625 .0588235294117647 .0 5 
.6
.4 .3 .2497 .2 .186a35 .16 .14 .1 2497 .1 .1 .0B .0 a35186 .0 9724 .09 .0857924b3643... .08 
1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) 
Bases and their compatibility with numbers
Intervals of 2:
Good: Octal Dozenal Okay: Senary Bad: Decimal 
Intervals of 3:
Good: Senary Dozenal Bad: Decimal Octal 
Tetradecimal

Hexadecimal

Octadecimal

1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) 
.7
.49 .37 .2b .2 49 .2 .1a7 .17ac63 .1 58 .13b65 .12 49 .1 .1 .0d .0c37 .0b75a9c4d268... .0 ac6317 
.8
.5 .4 .3 .2 a .249 .2 .1c7 .1 9 .1745d .1 5 .13b .1 249 .1 .1 .0f .0 e38 
.9
.6 .49 .3ae7 .3 .2a5 .249 .2 .1 e73a .1b834g69ed... .19 .16gb .1 52a .13 ae73 .1249 .1 .1 
1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) 
Bases and their compatibility with numbers
Intervals of 2:
Good: Hexadecimal Okay: Octadecimal Bad: Tetradecimal 
Intervals of 3:
Good: Octadecimal Bad: Hexadecimal Tetradecimal 
Chart Conclusion
As you can see, the dozenal and octadecimal systems seem to be the clear victors. With around half of the divisions coming out as short terminations, these bases prove to be very useful in most mathematical calculations.
However, the dozenal system wins in my opinion because of 3 factors:
1: You only need to learn 2 extra symbols in dozenal instead of the 8 new symbols required for octadecimal. Imagine all the names you'd have to remember for each new digit.
2: Dozenal has the advantage over all multiples of 2, whereas octadecimal only has advantages over multiples of 3. Since the multiples of 2 are far more useful, dozenal also wins this.
3: There are less occurrences of recurring decimals. For instance, in octadecimal, when you compare it to dozenal
it loses to 1/4's, 1/8's, 1/C's, 1/G's and only has a slight advantage over 1/E and 1/I.
Dozenal is simply more compatible with more numbers.
However, the dozenal system wins in my opinion because of 3 factors:
1: You only need to learn 2 extra symbols in dozenal instead of the 8 new symbols required for octadecimal. Imagine all the names you'd have to remember for each new digit.
2: Dozenal has the advantage over all multiples of 2, whereas octadecimal only has advantages over multiples of 3. Since the multiples of 2 are far more useful, dozenal also wins this.
3: There are less occurrences of recurring decimals. For instance, in octadecimal, when you compare it to dozenal
it loses to 1/4's, 1/8's, 1/C's, 1/G's and only has a slight advantage over 1/E and 1/I.
Dozenal is simply more compatible with more numbers.
Zero Enders
This is what each base looks like if we stop our multiples when we reach a 0 in the one's column. Since 0 resets the pattern, you can see how some bases are better for multiples of 3, and others are better for multiples of 2:
When we look at 12, we notice how multiples of 2 and 3 are wonderfully short. 2 and 3 are the most important primes, they are the most common divisors of all composite numbers (a whole number that can be made by multiplying other whole numbers).
Primes
This leads me to primes. Primes are when a number has no divisors other than 1 and itself. Primes are fixed in value, but they can be shuffled in notation (ie. what numbers we use to represent them), depending on which base you use. For instance, the prime number 13 that we know in decimal is now 11 in dozenal.
You are used to seeing primes in decimal like these:
You are used to seeing primes in decimal like these:
These are dozenal primes:
In each column, notice the gaps between primes where there are no primes.
For dozenal, you have no primes in:
 3 columns: {2, 3, 4} (excluding numbers 2 and 3)
 1 column: {6}
 3 columns: {8, 9, ᕍ}
8 columns are composite.
4 columns are mostly prime.
For decimal, you have no primes in:
 1 column: {2}
 3 columns: {4, 5, 6}
 1 column: {8}
6 columns are composite.
4 columns are mostly prime.
Imagine we take our number 2 and look at all the multiples towards infinity. This captures half of the numbers, all of the even numbers.
Now we do the same with the number 3, looking at all the multiples, we find that every second one is an even number.
We are left with 1/4 of all numbers that remain, only some of these numbers are primes.
For example, in decimal, 25 is not divisible by 2 or 3, but it is not prime, since it can be divided by 5. But, as a general rule of thumb, you can see that after using 2 and 3, we capture most of the composite numbers.
5 is prime, so it is the divisors of quite a few composite numbers that 2 and 3 miss. But, 2 and 3 are far more important than 5 and 7. This makes base 12 and 18 the clear winners again. However, because dozenal is better for multiples of 2 than octadecimal, this again suggests that dozenal is the best base ever to exist.
For dozenal, you have no primes in:
 3 columns: {2, 3, 4} (excluding numbers 2 and 3)
 1 column: {6}
 3 columns: {8, 9, ᕍ}
8 columns are composite.
4 columns are mostly prime.
For decimal, you have no primes in:
 1 column: {2}
 3 columns: {4, 5, 6}
 1 column: {8}
6 columns are composite.
4 columns are mostly prime.
Imagine we take our number 2 and look at all the multiples towards infinity. This captures half of the numbers, all of the even numbers.
Now we do the same with the number 3, looking at all the multiples, we find that every second one is an even number.
We are left with 1/4 of all numbers that remain, only some of these numbers are primes.
For example, in decimal, 25 is not divisible by 2 or 3, but it is not prime, since it can be divided by 5. But, as a general rule of thumb, you can see that after using 2 and 3, we capture most of the composite numbers.
5 is prime, so it is the divisors of quite a few composite numbers that 2 and 3 miss. But, 2 and 3 are far more important than 5 and 7. This makes base 12 and 18 the clear winners again. However, because dozenal is better for multiples of 2 than octadecimal, this again suggests that dozenal is the best base ever to exist.