I would imagine it to be something that has a few of the following characteristics:
I Googled this question and was surprised to see an article claiming it was a star? How does the roundness of this star compare to the event horizon of a black hole or a surface of a neutron star?
The Sun is a contender, based on its size and our measurements of it's sphericality (is that the proper word..?).
The sun is nearly the roundest object ever measured. If scaled to the size of a beach ball, it would be so round that the difference between the widest and narrow diameters would be much less than the width of a human hair.
From Roundness of the Sun and this question Why is the Sun so Spherical I asked a while back.
At the other end of the scale is the electron; (but is it really an "object", probably not.)
....and still found no signs of an electric dipole moment in the electron. The electron appears to be spherical to within 0.00000000000000000000000000001 centimeter, according to ACME’s results.
From Dipole Moment of Electron
I suggest these examples because we have data on them, as opposed to models.
Given that certain physicists are indulging in the idea of multiverses, which are universes that are causally disconnected from us and so unproveable to be shown to exist. I should point out that one of the very first multiverse idea was Platonism - where we had one universe of actual physical things, the things we see right in front of us and in the night sky, and the ideal universe of ideal forms.
If we take this as a plausible picture, and also expand our notion of universe to mean everything that there is, then we take in both these universes. The the most perfect circular thing is the idea or the form of the circle. Nothing can be more circular as it is the very idea of the circle.
I should point out that there is at least one physicist that takes neo-platonism seriously: Shimon Malin in Nature Loves to Hide: Quantum Physics and the nature of reality - A Western Perspective.
You could refer to Why is the Sun almost perfectly spherical?
Blockquote The relationship between oblateness/ellipticity and rotation rate is treated in some detail here for a uniform density, self-gravitating spheroid and the following analytic approximation is obtained for the ratio of equatorial to polar radius $$ \frac{r_e}{r_p} = \frac{1 + \epsilon/3}{1-2\epsilon/3}, $$ where $\epsilon$, the ellipticity is related to rotation and mass as $$\epsilon = \frac{5}{4}\frac{\Omega^2 a^3}{GM}$$ and $a$ is the mean radius, $\Omega$ the angular velocity.
Putting in the relevant numbers for the Sun (using the equatorial rotation period), I get $\epsilon=2.8\times10^{-5}$ and hence $r_e/r_p =1.000028$ or $r_e-r_p = \epsilon a = 19.5$ km. Thus this simple calculation gives the observed value to a small factor, but is only an approximation because (a) the Sun does not have a uniform density and (b) rotates differentially with latitude in its outer envelope.
Now you can see from the formula above that $\epsilon$ decreases with increasing mass, decreasing rotation rate and decreasing mean radius. Thus objects with stellar masses will become progressively more spherical if they have smaller radii and slow rotation.
Neutron stars have $a \sim 10$ km, and the fastest rotation have $\Omega \sim 1000$ rad s$^{-1}$. For a 1.4 solar mass neutron star this yields $r_e/r_p \sim 1.0067$, i.e. more oblate than the Sun.
However, the rapidly-rotating "pulsar" phase of a neutron star's life is comparatively brief. After a million years or so, the rotation rate has slowed by many orders of magnitude (to periods of a few seconds or more) and will continue to decrease through the emission of magnetic dipole radiation. Once the neutron star above has slowed to $\Omega <60$ rad s$^{-1}$ (i.e. a rotation period $>0.1$ s) then it will become more spherical than the Sun.
The Sun is not going to be the record holder as far as "normal", non-degenerate stars go. There are low-mass M-dwarfs with rotation periods of $\sim 100$ days and radii about 0.1 that of the Sun, that will have $\epsilon \sim 10^{-9}$. For a neutron star to beat this it would need to have a rotation period longer than about 20 seconds. Although such neutron stars have not been observed - because the pulsar mechanism switches off at periods longer than a few seconds, it seems likely that the majority of neutron stars in the galaxy rotate as slow or slower than this.
Thus my answer would be that the most spherical observed object would be something like Proxima Centauri with a rotation period of 83 days and about 0.15 times the radius of the Sun, the galaxy probably has several hundred million, slowly rotating old neutron stars that are more spherical than this.
