I made an Arduino based robotic arm that's less than 2 inches (5 cm) tall for under $50. But is it the smallest? The attached video will take you through the design, assembly, and code process. My goal was to make this arm as small as possible while still being able to be printed on a standard 3D Printer. I think this arm turned out pretty well!
Details
All info (including part links) can be found here if you wish to build this robot arm: https://youtu.be/6de55CUOTI4