How to give a champion 5 to 12 abilities.
There are already champions in the game who have more than 4 abilities. The president for these champion involves an ability that toggles the form of the champion, usually through their ultimate, which changes the form of their other abilities.

And some champions who have limited access to certain abilities, or have changed abilities based on conditions.

There may be a few others that could be mentioned, but I think you guys get the idea.
Well, what if a champion was at least partly based around this, but with a twist.
To start out small, the basic concept is that a champion would activate an ability by pressing one of the hotkey buttons, lets use Q as an example. After activating this ability, there could be some sort of active effect or at least a visual effect triggered, but more importantly W and E will now change to augment the effect of this ability. After one of these effects are chosen, all abilities revert to their previous form with Q on cooldown. Q would also be available to be pressed again to deactivate the ability, returning all of the abilities to their previous state.
This could be applied in several different ways with several different effects and augmented effects. I would say the best way to go about it would be to have all of the augmented effects be similar, or at least accomplish the same goal but in a slightly different way. For example, if Q is intended to be a damaging spell, by pressing Q the champion visually begins to draw in energy from around themselves. They then have the choice of pressing W or E. W will unleash the energy as a linear skillshot and E will release the energy in a nova.
A more defensive example could be that the augmented options determine if the champion creates a shield that blocks incoming physical or magic damage.
Ranking. I would recommend making the rank of each ability determine the effect of their respective augmented abilities. For example, 2 ranks in Q, then selecting the W augment will cause it to activate with it's respective rank 2 effects, the same with E, regardless of the number of ranks in any other ability. (This is admittedly worded very poorly, but I hope you can understand)
Now, to go crazy with this idea. A champion could have 4 basic abilities with no ultimate, much like Udyr, and with this concept activating Q would give 3 options W, E, and R. Or the player could activate their W and have 3 options, Q, E, and R. Or they activate their E and pick from Q, W, and R. And of course lastly, if they activate their R they pick from Q, W, and E. This adds up to 12 different abilities.
This is still a very high level concept and I have very few actual ideas for what would even be good abilities, aside from the examples I gave. At this point I'm looking to see if anyone even thinks this could make for an interesting champion, as well as some ideas for abilities and their respective augmented abilities.
If anyone also wants to take this idea and try and build it into your own champion concept, that is perfectly fine with me. I would ask that you comment with a link to your post so I can take a look at it.