If you come across a 17 button controller, odds are that 16 buttons are for directional and attack inputs, whereas the 17th button is a function button.
What more can you ask for? The KBGM-H09 has a whopping 17 buttons and is suitable for both desk & lap play!
Perhaps the most interesting aspect about the Sunga G1 stickless controller is that it has five more buttons than what we are used to.
The Titan is loosely based on the Sega Saturn controller, and one of its many unique features is having a customizable encoder knob.
Ultra Arcade's Standard series is a reasonable entry point into its customizable controllers, and there are Pro & Lite editions available.
The Mixbox comes in variations from all-keys to southpaw, but you may be most interested in its S+ variant that has 13 buttons and a spacebar.