Hey Countingman-
Hexadecimal is base-16, I think what you want is tetradecimal, aka base-14. Counting in a base higher than ten will get easier with practice. You just need to assign new symbols for the numbers ten, eleven, twelve, and thirteen, and the standard choice is A, B, C, and D. The units are no longer ten, one hundred, one thousand, etc, but now fourteen, fourteen squared (196 base-10 = 100 base-14), fourteen cubed (2744 base-10 = 1000 base-14), etc. Counting up to 196 in tetradecimal is
1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, 10
11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 20
21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B, 2C, 2D, 30...
...
91, 92, 93, 94, 95, 96, 97, 98, 99, 9A, 9B, 9C, 9D, A0
A1, A2, A3, A4, A5, A6, A7, A8, A9, AA, AB, AC, AD, B0
B1, B2, B3, B4, B5, B6, B7, B8, B9, BA, BB, BC, BD, C0
C1, C2, C3, C4, C5, C6, C7, C8, C9, CA, CB, CC, CD, D0
D1, D2, D3, D4, D5, D6, D7, D8, D9, DA, DB, DC, DD, 100
If you want to do this "vortex math" thing about doubling and adding the digits in base-14, it goes like this:
1
2*1 = 2
2*2 = 4
2*4 = 8
2*8 = 16 in base-10 = 12 in base-14, digit sum of 12 = 3
2*3 = 6
2*6 = 12 in base-10 = C in base-14
2*C = 24 in base-10 = 1A in base-14, digit sum of 1A = B
2*B = 22 in base-10 = 18 in base-14, digit sum of 18 = 9
2*9 = 18 in base-10 = 14 in base-14, digit sum of 14 = 5
2*5 = 10 in base-10 = A in base-14
2*A = 20 in base-10 = 16 in base-14, digit sum of 16 = 7
2*7 = 14 in base-10 = 10 in base-14, digit sum of 10 = 1
So you get the cyclic pattern 1, 2, 4, 8, 3, 6, C, B, 9, 5, A, 7 which cycles over twelve numbers. You can draw the flow diagram on a dodecagon and it will make another "demented butterfly" pattern like the base-12 example.
A good rule of thumb to get a nice full flow diagram that hits lots of points is to choose a base N such that N-1 is a prime number. This will ensure that your flow diagram hits N-2 points. For example, if you choose base-20, 20-1 = 19 which is prime, so then your diagram will hit 18 points.
We aren't restricted to only doubling the numbers. You might get some pretty interesting cycles if you do other operations like tripling or quintupling and then adding the digits. I haven't really investigated much of this "vortex math" in alternate bases except to explore db101's comment a little further. Once you get the hang of counting in an alternate base you can probably have a lot of fun with it.