It has just been proven that only 15 types of convex pentagons exist that can tile the plane. Which implies that there just might be a single polygon (almost undoubtedly concave) that can tile the plane in a non-periodic manner (as do Penrose tiles; but PTs require two different figures, not a single figure).

(If you’ve ever played with regular pentagons, you have discovered that they can’t tile the plane without gaps or overlapping. The pentagons referred to in this proof are NOT regular. Here is one such example, taken from the article:)

You can see many of the details at the following link.