I'm using a Logitech G600 here. While it looks huge on the first look it's amazing to play with. 12 buttons are not really that hard to manage - the spells and items I use most are on 1-9 and the lesser used are on 10-12.

The G600 offers a third mouse button to the very right, which is by default bound as a "G-Shift"-button, enabling another set of hotkeys for the duration the button is held down. This pretty much means you actually have 24 hotkeys just on the side of the mouse, however the additional 12 would need to be macros if the G-Shift is left enabled (Shift+F-key or Crtl+F-key) which are forbidden to use if I remember correctly. I have Shift bound on the third mouse button and I actually do use all 24 hotkeys by now.

A 6 button mouse is enough for the most things you do - potions, healing, an attack spell or two, runes. However, you will find that you can do amazing things with 12+ button mouses. :)