Why is there no large asymmetrical heavenly body then? Or are there?
Once things get big enough, the force of gravity alone is enough to squish them into a ball. In fact, being big enough for this to happen is one of the conditions of something being defined as a planet by the IAU (the other conditions being that it orbits around a star and that it has also cleared all other large objects out of its neighborhood).
There are a bunch of asymmetrical asteroids. That’s about as big as you can get and remain asymmetrical.
Htrae am not round.
I do fancy them hip bones on my heavenly bodies …
The dwarf planet Haumeais shaped like an egg.
Which is the controversial bit, since Earth hasalso not cleared its neighbourhood,according to many.
Well, not exactly egg-shaped like what’s in the fridge, but rather ellipsoidal. Though not an ellipsoid of revolution it still has a certain symmetry to it, as the OP was asking. And come to think it, just to wreck my own line of reasoning, eggs be symmetric after all. 
The compressive strength of naturally occurring materials is limited, so you can only have a column of it be so tall before the rock at the bottom is overloaded and crumbles/extrudes out from under it. Take granite, for example, with a compressive strength of 29,000 psi and a density of 0.0991 pounds per cubic inch; under earth’s surface gravity, the tallest exposed cliff face you could make would be 4.62 miles tall; any taller than that, and the rock at the bottom is overloaded and will crumble. This is actually something that happens in very deep mines: a rock burst occurs when highly compressed rock is exposed by miners, and it violently explodes toward them.
One way around this limit is gigantic cones, so that the load from the uppermost mass is distributed across a much broader base, i.e. mountains. But here on earth (and I expect on other planets of similar/larger size) even this is bounded because the massive weight of a mountain depresses the crust beneath it so that the summit is not as high above the planet’s mean surface as you would otherwise expect. Even the weight of a glacier is enough to push the earth’s crust down: the shorelines of the Great Lakes and Hudson bay are slowly shifting as the earth’s crust in that region rebounds after having been depressed by the glaciers that used to be there many thousands of years ago.
The larger the planet, the stronger the surface gravity - and so the less will be the maximum surface irregularly you would expect to see. The smaller a planet - down to the point of not even being regarded as a planet anymore - the greater the out-of-roundness that can be supported by the material composing it. In the extreme, I’d guess that a column of granite in space with a 1-square-mile cross section could be many, many hundreds of miles tall (especially if you tapered each end to a point, i.e. two cones attached base-to-base) before it would threaten to collapse under its own gravity. Something like that would represent the greatest possible degree of asphericity.
Saturn spins fast enough that it is visibly ellipsoidal, but that’s just the atmosphere. I’m pretty sure the rocky core is round.
I thought there was still some debate about whether or not Saturn has a solid core.
Just give it time.
As is often the case, Feynman gives an explanation.
Any sufficiently large spinning celestial body is going to take the shape of an oblate spheroid. The earth itself is 26.6 miles thicker across the equator than it is pole-to-pole due to its spin.
I had the impression that the OP was interested in deviations from the oblate-spheroid shape due to local geological surface irregularities such as mountains and valleys (as opposed to whole-planet distortion due to the effects of rotation and/or tidal forces).
Note that it is not known that Haumea is a smooth egg shape. The shape shown in the diagrams there is just a best-fit ellipsoid. There’s always some ellipsoid that fits better than other ellipsoids, but that doesn’t mean that any ellipsoid at all is a particularly good fit.
Vesta is a fairly large object that’s not spherical. It’s close to the size limit for that. Anything much larger would recircularize itself.
Vesta wasn’t always non-spherical. It started off as a sphere, but that was when its interior was hotter and softer. But it cooled off and the rock got stiffer. Then it suffered two big collisions that by coincidence hit in the same place, which is now at the south pole. Those collisions removed about 1% of Vesta’s mass and made it distinctly out of round.
I needed the term “asphericity” for my OP but didn’t know it yet.
I guess I’m interested in the evolution of a planet from birth and whether it’s spherical from that moment or it becomes that, and why/how. Not really geographical features, but why a large chunk would be virtually round.
Large chunks are round because of gravity. Any part that sticks up above the sphere gets pulled back by gravity.
Small chunks often aren’t round, because they are small they have very weak gravity, and therefore the force of their own gravity isn’t strong enough to crush them into spheres.
There are two competing forces here. The structural forces that hold solid objects together, and the force of gravity. Imagine a square block of granite on a field. It has a certain structural strength. And there is a force pulling down on that block of granite. If the block of granite is twice as big, it’s volume is 8 times greater, which means the force of gravity acting on it is 8 times greater, but the structural forces holding it together are exactly the same. As the stone block gets bigger and bigger eventually gravity will crush it. Or alternatively, it will sink into the ground below. The solid ground beneath your feet only has so much strength to hold up so much weight.
And so there’s a limit to how big a mountain can be before it crushes itself or sinks into the crust. And that limit depends on the structural strength of the material the mountain is made of, the strength of the crust under the mountain, and the strength of gravity on whatever planet the mountain is on. A mountain 10 times bigger than Mt Everest would collapse under its own weight, and also sink into the mantle of the Earth.
So you can think of any non-spherical parts of a planet or moon as giant mountains sticking up. What are the forces acting on the mountain to pull it down?
So in our Solar System, the asteroid Vesta which is about 500 km across is noticeably non-spherical. But it’s pretty close to spherical. Smaller asteroids and moons can be even more aspherical, larger bodies are more spherical.
Larry Niven’s “Jinx” is an egg shaped moon more massive than earth. It was originally shaped that way by tidal forces from its primary, and got ‘stuck’ that way after it cooled.
In the past, I had pretty much accepted this as reasonable (Niven is a smart guy, after all). Now that I actually think about it, it seems impossible for it not to return to spherical after leaving the region of tidal stress, especially given it’s extra large mass. Still, it makes a nice story.
When The Stranger smacked down the Over Mind in the “Fantastic Four” comic, The Stranger divulged that The Stranger came from a planet. . .
That dwarfed galaxies!
Whoa!
I recall from a few years ago this interesting article speculating on the theoretical possibility of toroidal *(aka *“donut shaped”) planets. They’d pretty much have to be artificially created though.
OK, I thought you were going to link to my old Staff Report on the topic, but that article is far more detailed. Bravo, to the author of it.