How did the universe get so big so fast? 
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*The universe started at the big bang around 15 billion years ago.

*The universe is now at least 92 billion light-years in diameter.


Together, don't these mean that the universe, at some time in the past or present, was expanding at a rate of faster than 1 ly/y in every direction? Why is that possible?
 A: Alright, here's the skinny. The universe is much much larger than 92 billion light years. But the region we can see is around that number. Also, you have it all wrong for the expansion rate of the universe. The expansion is more like stretching. No matter where you are (that's an approximation), you would see a length x expand by the same amount after a time t. You seem to be thinking of it like there's some border expanding away from us, when really it's the space everywhere that's just getting larger. This is a cumulative effect; the farther away things are, the faster they seem to expand away from us because there is more space in between to be stretched.
The current rate of expansion is called the Hubble Rate. It is estimated at around 500 km/s/Mpc (Mpc is megaparsec). Or, alternatively, 160km/s per million light years. That means that every second a distance that was $1000000ly$ becomes a distance of $(1000000+1.691\times10^{-11})ly$. Now, you say the observable universe is 92 billion light years across. Then the farthest reaches are 46 billion light years away. This means that every second, somewhere around 7.4 million kilometers are added to that space. Technically, that means that objects at the edge of the visible universe are receding faster than the speed of light, but they are not moving faster than light.
If I were moving faster than light, you could stand next to me as I pass and it would look like I'm going faster than light. However, these objects at the edge of the observable universe are not moving faster than light. If I positioned myself next to them, they would not be moving at all. Their motion with respect to space, to me, to anything near them is not above the speed of light, it is simply that more distance is added between them and us each second than light could traverse.
Place a beam of light at that distance travelling away from us. To us, the beam of light would be travelling at $c$ plus the speed due to expansion. So no matter what, the objects out that far are not moving faster than light.
The point here is twofold. First, the expansion rate is not a definite speed; it is a speed per set distance length. And second, nothing is actually moving faster than light ever; space makes things recede faster than light, but space isn't really moving either.
