The Solar System is a beautiful thing to behold. Between its four terrestrial planets, four gas giants, multiple minor planets composed of ice and rock, and countless moons and smaller objects, there is simply no shortage of things to study and be captivated by.

There is many other planet like Saturn. And, apart from the ring that is part of the planet but technically it is not attached to it, so it doesn't really count, you can clearly see it is not round in a mathematical way. I mean, it is not a sphere but an oblong shape. Really all planets in our Solar System are oblong but Saturn is the most noticeable to the naked eye. Earth is oblong too but the difference between the polar radius and the equatorial radius is so minimal, it is impossible for the naked eye to tell the difference from a perfect sphere. But I suppose you were thinking about more exotic shapes. Things like doughnuts, pills, discs or just potatoes.

Well, the planets are spherical or oblong due to gravity. Planets grow by the constant aggregation of matter attracted by gravity or just because an errant object collides with the forming planet. When a meteor collides with Earth, it either burns completely in the atmosphere or hits the ground leaving a crater. In either case, the matter from the meteor is added to the mass of Earth. This process in the early times of the Solar System formation was much more frequent because there were much more objects floating around. Objects that also were created by the aggregation of minor objects.

The bigger the object gets, the bigger the mass and hence the greater the effect of gravity on it. Gravity tend to give planets a round shape because it is the shape where all points from the surface are at the same distance from the center of gravity. This is called hydrostatic equilibrium. A dice shaped planet would be possible but meteors impacting on the surface would tend to erode the borders and corners of the cube and the expelled matter would tend to accumulate in the middle of the sides (the corners would be like giant mountains and any loose rocks would roll down to the center of the sides). So it would end up being round shaped anyway. It is similar to what happens with round stones at the sea shore. They start as pointy irregular rocks put erosion gives them rounded shapes after many years.

But then why Saturn is oblong and not perfectly spherical? (And all the other planets in our Solar System, for the matter) this is because gravity is not the only force that acts on the matter of the planet. There is also the effect of centrifugal forces due to the spinning of the planet. This tends to throw matter away from the axis of rotation (the polar axis) so the matter around the equatorial zone tends to escape giving the sphere its oblong deviation. Saturn spins fast and it is mainly composed of low density gas, so the effect is greater on it than in other planets.

Mercury is the most rounded one. It is a rocky planet with an inside more solid than the inside of Earth so the effect of centrifugal forces is no so strong on it. 

Earth is also a rocky planet but the inside of it is molten material, mainly iron so the solid crust is really a thin layer and the centrifugal forces can stretch it a bit.

So, spheres and oblong spheres are the most likely shapes for planets, either small rocky ones or big gaseous ones but can they have other shapes?

This most probably can happen in smaller rocky ones where the pull of gravity is not strong enough to model the shape of the rock. Asteroids and planetoids of our own Solar System are good indications of what we could find out there in other Solar Systems.