There seems to be a standard set of 8 inks (9 including photo and matte black) that everybody uses (CcMmY plus black and 2 grays). Epson generally puts that in 8 channels by switching PK and MK,while Canon and HP use 9 channels to avoid the black switch.
At 10-12 channels, approaches differ a bit.
Epson most commonly adds orange and green to their 8 ink system, to get 10. Variants include losing a gray and substituting violet, and also starting with 8 (forgoing orange and green), but ADDING a gray and eliminating the PK/MK switch. I am not aware of any Epson printer with more than 10 channels, although there may be one.
Canon and HP start with 9 channels (no PK/MK switch) and add different things to reach 12. Canon's longtime approach has been to add red, green and blue, and HP loses cyan (while retaining light cyan) to add red, green, blue and a gloss optimizer. Recent Canons sacrifice green for a gloss optimizer, and, in a few desktop printers, also sacrifice blue for an extra gray (still 12 channels in both cases).
Assuming that Epson's violet is a relative of what Canon and HP call blue, and that Epson's orange is related to the Canon/HP version of red, we have a total of 14 possible channels used in current printers
3 bluish colors:Cyan/Light Cyan/Blue or Violet
3 red/magenta colors: Magenta/Light Magenta/Red or Orange
5 Neutrals (some third party ink sets use even more, but I've never seen a printer manufacturer use more than 5): PK/MK/light gray/gray/dark gray
1 Yellow
1 Green
1 Gloss Optimizer
Will anyone make a 14-color printer? If someone were to make a 15 or 16 color printer, what would the extras be? An extra dilution of yellow or green? 6 or 7 neutrals (there is a Cone ink set with 7, I believe)? Metallic colors aimed at the graphic arts market?
At least a 14-color printer seems to offer real advantages... Start with Epson's 10 color inkset , which has the best gamut around except in the deep blue/violet (where Canon's and Hp's blue inks have real advantages), then add a dark blue ink, an extra gray, a gloss optimizer and eliminate the PK/MK switch - that's 14 colors, and Epson has actually used them all (except the gloss optimizer), just not in the same printer.
Going all the way to 16 channels, at least one of the extras would be a gray (for photographers - the graphic arts market might prefer to use them on metallics, or a white ink to print on black media) - is the other a seventh gray, or light yellow-green?