Does gravity stack? If I am standing on a planet on the opposite side from its sun will I feel the downward pull of both the planet's gravity and the star's gravity?  In other words, does gravity "stack" (Any D&D players out there?)
 A: Technically, yes, however in the case of you standing on a planet:
You feel the gravity of the planet and star, so you accelerate accordingly.
However the planet also feels the gravity of the star, so accelerates towards it. So you only notice your acceleration towards the planet.
A: Well given Newtons famous equation the force of gravity you feel is equal to
$$F_{\textrm{gravity}}=G\frac{mm'}{r^2}.$$
It increases with the mass of the two objects being measured and is inversely proportional to the distance between the two objects in question. Strangely enough however, this technically stretches out nearly infinitely so the gravitational force of a black hole a billion light years away is technically exerting an almost completely negligible but still (hardly) measurable force upon you right this instant. So yes, gravity does "stack" but in a more equilibrium fashion. Large masses you are closer to exert more force than those farther away.
A: It might not be best to say you'd feel it, as the oscillations of the sun's gravity (as due to rotations and revolutions) are relatively subtle here on earth. The moon's gravity has twice as much influence in terms of tidal force, which is probably the most prominent consequence of gravitational "stacking" in the macroscopic sense you had in mind.* Here's an illustration of a planet's tidal field by Krishnavedala:



In the case of the earth and the moon, the oceans are pulled toward and away from the moon, causing high tides directly below the moon and on the opposite side of the earth. Meanwhile, low tides occur at points halfway across the surface of the earth from where the moon hangs directly overhead. Other consequences of tidal forces include the preservation of some planetary rings (i.e., prevents them from coalescing into moons) and the splitting of Shoemaker–Levy 9 before it hit Jupiter. These suggest other scenarios in which tidal forces could be well beyond perceptible, and even deadly.

* In a more microscopic sense, which I think @pfnuesel may have had in mind, gravity would not be perceptible at all if it didn't "stack". If only the single subatomic particle nearest you had any gravitational effect and the rest of the earth didn't, none of us would be here!
