Q:

What would be the conditions and causes that would make a planet not spherical?

Introduction
I have been looking for information on the Internet. I have discovered that the shape of the planets is due to the gravity to which they are subjected, so they are spheres (,but they are not perfect spheres). This also has to do with the sphere: the sphere is the form of least energy and nature favors these forms.
I have also read different posts on the subject. So if these different forms existed I could not find a good explanation of why such a body would be generated or if there could be, for example, a system full of these "planets".
For example this post explains the possibility of having a planet with a cube form.
And this other one clarifies the existence of these types of nonspherical planets, but does not discuss its origin in depth.
Question
So, under these meager knowledge of this theme, I asked myself the question: what would be the conditions and causes that would make a planet not spherical?
For the answer
I would like the answer also to resolve the doubts that stem from this, for example: Does it depend on the gravity of a single body? Do different magnetic fields intervene?
I do not put limits on "who" or "what" creates the causes or the origin of the form of the planet, but I look for an explanation other than "magic".
How long does this planet have to survive? Long; take the earth as a reference.
At what point do you define your planet as not spherical? You can at first sight see that it is not like a circle, like an ellipse or similar. Based on this Jupiter has a spherical body.
By "planet", do you mean the same as NASA or is, for example, a moon like Phobos a possible answer?

It is massive enough to be rounded by its own gravity.
It is not massive enough to cause thermonuclear fusion.
It has cleared its neighbouring region of planetesimals.

A:

Based off this question on physics SE a dumbell shaped planet is possible if the rotation rate is high enough. Though to speed it up too much results in the planet breaking apart (as seen below).

What you want to avoid is reaching the planet's mass shedding limit - the point at which the centrifugal forces are greater than the gravitational attraction and the planet separates. Given the right mass and rotation rate it isn't impossible that a planet this shape could form.

A:

As far as I can see you've kinda shot yourself in the foot with:

By "planet", do you mean the same as NASA or is, for example, a moon like Phobos a possible answer?

It is massive enough to be rounded by its own gravity.

You've basically told us it is a sphere and its gravity is strong enough to pull it back into a spherical shape so whatever someone comes up with can't be stable.
That being said you can have something that orbits quickly - and so becomes an ellipsoid (though you said Jupiter doesn't count and that is an ellipsoid). In an extreme this would become a disk...but the centrifugal forces would probably tear the outer bits of rock off if it went that fast.
Another possibility is an early collision (when the planet is hot) with something very fast and tunnel right through from pole to pole. This would give you a doughnut sort of planet and, providing it isn't totally molten, this might be able to hold itself together - torus shaped planets are apparently stable.
So you could have a doughnut shape but I think its unlikely (you might want to ask a second question on just how much energy you would need to punch a hole through a planet for the size you want to get a handle on the likelihood of that).

