This was brought up in a discussion about sci-fi weapons, where people were discussing simply using kinetic projectiles from orbit to hit with tremendous force. That's obviously a theoretically possible thing. Where I'm being unsure is that some posters were claiming that you could get substantially more energy by launching a kinetic projectile just slightly slower than orbital speed, which would cause it to loop repeatedly around the planet (we'll say the Earth here) gaining speed on each pass until eventually it reaches the ground with enormous kinetic energy.
Okay, I'm not an expert in any of these fields, but I've read fairly widely (if shallowly) across a few relevant topics. So I'll offer some comments, fully expecting that experts will pull me up at some point.
Sure, kinetic energy weapons are a thing (ordinary bullets, for example, but also anti-tank projectiles such as tungsten or depleted uranium darts within armour-piercing, fin-stabilised, discarding sabot rounds), and the idea of kinetic weapons being used against a planet is plausible. But the ideas presented by the posters are veering way off course in terms of physics.
At a glance, "OK, it keeps falling toward the planet being accelerated by Earth's gravity so it goes faster and faster" makes sense, but...I'm pretty sure it doesn't work actually work that way. But my knowledge of physics and orbit dynamics is minimal and I've forgotten most of it. So I figured I'd crowdsource to a place with experts.
Let's have a look.
Some things that bother me about this:
* "It keeps gaining speed over a prolonged period of time" is possible with gravity involved, but always kind of a general physics red flag. Things go slower over time, not faster.
Objects gain speed as they fall into a gravity well, that is, towards the ground. If they move away from the ground, then they lose speed. To look at objects in a stable orbit: if the orbit is perfectly circular then the speed doesn't change; but if the orbit is elliptical then the speed increases as the object moves towards the lowest point of its orbit, then decreases as it moves towards the highest point.
The second thing is that orbital speed is directly related to altitude: the lower the orbit the higher the orbital speed. Thus the International Space Station at ~400 km altitude takes about 90 minutes to complete an orbit, a geosynchronous satellite at ~35,800 km altitude takes a day to complete an orbit, and the Moon at ~380,000 km takes about a month to complete an orbit. Because of this strict relationship, if an object's speed is reduced at some point in its orbit it will lose altitude (seemingly paradoxically, gaining speed as it dives, as pointed out above); if the loss of speed is sufficient, the object will hit the Earth.
So the idea that an object circling the Earth can keep gaining speed is wrong. It can keep gaining speed only by continuously firing some sort of rocket engine. And the problem with gaining speed is that this will force the object to gain altitude; this is exactly how spacecraft are placed into higher orbits.
The one way for an object to gain speed as it dives towards the Earth (that is, to gain speed beyond what I've described above) is for it first to be placed in an orbit which will intersect the surface of the Earth (that is, it's definitely going to collide), and then second, to fire a rocket engine to increase the speed at which it's diving towards the Earth. NASA did something like that with the Apollo 4 and Apollo 6 missions (I recommend reading the Wikipedia article on Apollo 4).
And while I think about it, it might be worth pointing out that the idea you quoted in the first paragraph, of having an object travel around the Earth at
less than orbital speed so it will keep looping around the Earth and
gain speed...makes absolutely no sense. It's sort of like saying that you can make a plane go faster by making it stall, so if you stall it enough times you can make it go supersonic.
* Orbits degrade over time; they get slower, not faster. This isn't an orbit, but...it is very similar.
Orbits degrade over time, but for very specific reasons. In the case of the Earth, the orbits of objects in low Earth orbit degrade because the objects are being slowed by the Earth's atmosphere. If nothing is done about it, then yes, those objects will eventually fall deep enough into the Earth's atmosphere that they'll fall to the ground. Objects in higher orbits will have their orbits affected by the gravity of the Sun and the Moon; eventually this could cause them to be moved into an elliptical orbit which hits enough atmosphere to cause them to fall to the ground.
Finally, an object travelling at less than orbital speed will fall back to the ground. This isn't just because of the Earth's atmosphere, but simply because the object's speed is too low. Think of, for example, the first two Mercury flights, of Shepard and Grissom - the rockets launching them could not reach orbital speed, so the spacecraft simply fell back to Earth. (What they didn't do was "loop repeatedly around the planet gaining speed on each pass"!).
* As far as I know, satellites and things HAVE degraded and dropped out of of orbit unintentionally, which should follow pretty much the same path as an object fired just below the speed of a stable orbit. They didn't cause massive city-destroying damage when they fell. 
You're right. For a few reasons. First, these sorts of objects aren't designed to survive re-entering the Earth's atmosphere, so they disintegrate, and most of their energy is dumped into the Earth's atmosphere. Second, these objects aren't dense, so if they survive atmospheric heating they'll be slowed down to a relatively low terminal velocity by the Earth's atmosphere, and their impact won't have much energy to spend on the target. Third, these objects don't have much mass, which is the other part of the equation relating to energy.
* If an object every got going faster than escape speed, it would just fly away from the planet and no longer be orbiting, so it appears to me that the absolute fastest a projectile that looped the planet could ever go would be just below escape speed for whatever altitude started at.
Well, the direction the object is travelling matters as well, along with the direction of any acceleration applied to it. This is where understanding orbital mechanics is useful. It's not intuitive, but it is reasonably straightforward. (And as a writer, I find such "non-intuitive but straightforward" concepts great for stories - readers who aren't familiar with such concepts will go "whoa, what?" and readers who are familiar with them will go "yeah!".)
* I have no idea how air resistance would factor in. Would a direct path have less air resistance than a looping one?
Oh yes, air resistance will matter. Remember that air resistance is used to slow down spacecraft coming to land on any planet with an atmosphere, such as the Shuttle coming to land on the Earth, or the various spacecraft which have landed on Mars. The path matters to an extent, but this is one area I'm not sure about. My first thought is that the amount of heating the object will experience is simply related to its speed when it hits the atmosphere - higher speed = more heating. The path matters to the extent that a shallow descent means the object will stay high in the atmosphere for longer, higher altitude means thinner atmosphere, thinner atmosphere means less heating; so a shallow descent means the object won't heat quickly, so if it has some means to radiate heat, I think a shallow descent means it won't heat up as much for a given amount of deceleration.
Heating matters because if the object heats up enough it will just disintegrate and its energy simply gets spent heating up the Earth's atmosphere. I've read (and I don't know how accurate this is) that heating is the main problem for the development of hypersonic cruise missiles the Russians have been making threats about - they move so fast through the atmosphere that they just melt.
And the other point already mentioned is that if the object isn't dense, whatever survives the atmospheric heating will be slowed down towards its terminal velocity.
Any thoughts on this?
The main takeaway from all this is that designing pure kinetic weapons is hard if you're trying to be targeted.
That is, if you want to devastate the whole Earth, then the simplest kinetic weapon is a redirected asteroid or comet that's more than 5 kilometres across (which is obviously a lot larger than a rocket-launched satellite): whether it explodes in the atmosphere or on impact, it's going to dump a shed-load of energy into the atmosphere, just like the one which killed off the dinosaurs.
OTOH, if you want to devastate a particular country or location, then you need objects which will definitely hit the ground, and do so travelling at high speed. This suggests a long and narrow dart-like object made of something very dense, and with some sort of thermal protection so it doesn't melt from passing through the Earth's atmosphere at high speed.
Frankly, it really sounds like nuclear weapons would be a better option, because they pack a lot of energy into a small package which can be delivered at any speed, because they don't rely on either speed or mass to cause damage.
