If an asteroid is in an elliptical orbit about the Sun with the Sun almost at the center of the ellipse what would the eccentricity of the ellipse be?

All natural orbits are ellipses. We can force an artificial satellite into a spherical orbit, but it won't STAY there without occasional adjustments.

The "primary body" - in this case, the Sun - is at one of the two focuses (foci) of the orbit. If the focus is very close to the "center" of the ellipse, then the eccentricity of the orbit (how much it varies from a perfect circle) is close to zero.