If an astronomer claims to have discovered an object with a very eccentric orbit, what best describes the orbital trajectory of the object?

It would look like a squashed oval.

An celestial object's orbital eccentricity is a dimensionless factor which specifies the extent through which its trajectory about another body detracts from a complete circle. A measure of 0 is a circular orbit, numbers between 0 and 1 generate an elliptic orbit, a parabolic escaping orbit and a hyperbola for value greater than 1. For values far greater than one it looks like a squashed oval.