A cone, possibly.
A conic section.
Because the base of a cone is round, it is the only three dimensional figure with only one vertex.
Yes, the point on the cone is a vertex.Typical math textbooks define a cone as "A solid figure with one circular face and one vertex." So yes, there is one vertex.
A hexagonal pyramid.
A cone+++The original answer also had "Cube", which is obviously wrong!
It could be a pyramid shape