This would have to be the most pointless exercise and I have never seen a round display, but if you just want the intellectual exercise, you could surely do it with an OLED display like the SSD1306 display, and they are dirt cheap. It wouldn't be the first time a round dial has been in a rectangular bezel and, who knows? you could even have the length of the hands vary in length to reach into the corners - something Cartier never imagined.
Thank you all for the responses, my hope was that after it I could add a button or two in it and make some animations, this would be something to entertain myself on free time and maybe give it to my daughter. I really hate rectangular watches, but if there's no other way. I was checking some round OLED displays on the internet and thought it could be done.
Palad1N:
I was checking some round OLED displays on the internet and thought it could be done.
If you have found a round OLED display, I would imagine it was made specifically for this purpose, and you should be fine with it. I don't think the drawing commands would be any different from a rectangular display, and it may use common libraries.
For all that, I submit you will better off with a rectangular display, as it is likely to be better suited for some more useful purpose later.
A quick Google search for "round oled display arduino" turned up several styles from 1.5" 128x128 for under $7 to a nice 1.39" 400x400 AMOLED you can get for about $30.
Pick one and look at the documentation to make sure they point to an Arduino library and provide sample code.