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A rock sitting on land, the ocean floor, or floating in space maintains its shape somehow. Gravity isn't keeping it together because it is too small, so I'm assuming it is chemical or nuclear bonds keeping it together as a solid. If not it would simply crumble apart. So, what type of energy maintains the shape of a rock, where did this energy come from, and is it slowly dissipating?

As a corollary, if a large rock is placed on top of a small rock, is the energy required to maintain the shape of the small rock 'used' at a greater rate?

No, the exact opposite is true.

The molecules in a rock don't stay together because they're spending energy. They stay together because of attractive chemical bonds. The molecules have lower energy when they're together than when they're not, so you have to spend energy to break the rock apart, not to keep it together.

There are various mechanisms that keep solid things together, they all have one thing in common: They reduce energy to a minimum! When you want to break it apart, it costs you energy to do so!

Examples of bonds are:

Hydrogen-Bonds, which are very weak and come from an asymmetry of the electron around the proton, in such a way that it is energetically favourable to form bonds instead of repel each other.

Ion-bonds, which can be quite strong, but the materials are often recalcitrant (brittle). Materials having ion-bonds are not pure, they are a mixture of two different elements, one positively charged, another negatively charged and they form molecules together, mainly due to the Coulomb force.

There are many more!

The amount of work done is equal to the distance moved times the force in the direction of motion. As the rock is staying the same shape it does not need to exert energy.

You may be thinking that the rock needs to expend energy in order to hold up its heavy mass in the same way our muscles do if we hold up a heavy weight. But muscles need to contract to lift a heavy weight and this requires continuous activity at the cellular level as explained in the answer to this question.

Consider an answer by contradiction:

Imagine the rock is in the vacuum of outer space with no energy able to be added to it.
Suppose it does use energy to maintain shape. Then at some point, it will run out of energy and the shape will change. Now, since it is out of energy and can't change shape, isn't it now maintaining shape without energy?

Once there was a paradox in physics: why electron doesn't fall "down onto" atom's kernel and they decided it was "rotating". But "rotating" it would emit radiation and thus would loose energy. So later then they decided that "rotating" electron looses no energy staying on its "enabled orbits" and only would emit radiation when changing orbits.

The rock is a "set" of atoms, actually. You should look into the root of things.

Gravity isn't keeping it together because it is too small

Gravity is huge on small distances (look up the canonical formula — it's having /R^2, actually). But it's being compensated by other forces. We might continue this for long, but obviously there's no reason to repeat well known sources: https://en.wikipedia.org/wiki/Fundamental_interaction

Back to your q-n: rock is a collection of atoms. Atoms poses colossal energy indeed but as electrons on theirs orbits don't loose theirs energy and there's no intensive radioactive decay occurring it's all being "more or less" balanced. And when it's balanced — you guess it.