Does any circular path not have a beginning in time as well?
If a circular path somehow arises in an instant then that instant was the beginning.
I don't think you have pointed to anything real without a beginning.
Sure an imaginary enclosed figure of any kind, circle, triangle, square, etc. does not have a beginning.
The beginning of a path is not the part that was made first! It's entirely possible to build a path starting at its terminus, or somewhere in the middle, and finish with the part people are supposed to start from. Even if the Camino de Santiago was built starting at its destination, the beginning of the path would still be in Saint Jean Pied de Port. It wouldn't matter that its origin in time were somewhere else.
Also, there are plenty of paths that have two ends, but neither one can be shown to be the beginning or the ending. There's a path from behind my apartment through the back woods that empties out into a clearing near the water. Does the path begin at my apartment, or does it begin at the water and end behind my apartment? There's no answer to the question that is true for all who traverse it.
Of course, this is just the spatial sense of the word "beginning", and I'm pretty sure the OP was talking about time.