Depending on how you want to define vertices, a circle either has no vertices or the circle is composed of infinite vertices. A vertex is where two straight edges meet. Since a circle is continuously curved you can easily argue that it has no straight edges to meet and form a vertex. From a limits standpoint though it has infinitely many straight segments - but they are are all of length 1/∞
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The shape that has no vertices and is round is a circle. The circle is the only shape that does not have vertices.
Zero vertices
None but it's possible to construct shapes within a circle that have vertices.
Inscribed
Circle or sphere.