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In short, Neptune's gravity affects Earth far less than Mercury.

In Depth:

The effects of gravity depend on two factors: the mass of the object, and the distance to the object.

Assuming your question is limited to the planets within out own solar system, the answer would be a contest between Neptune (the farthest away) and Mercury (the least massive).

According to NASA websites, Neptune is approximately 309 times more massive than Mercury.

Determining distances is a much more complex problem, since both planets and Earth are in an elliptical orbit in which they vary their distance from the sun. In addition, they don't spend equal amounts of time at all distances.

Using the numbers given by other WikiAnswer questions, Neptune is an average of 30 AUs from Earth and Mercury is an average of 0.6 AUs from Earth. Which means Neptune is an average of 50 times more distant from Earth than Mercury.

Because of the inverse square law of gravity, Neptune affects Earth 2500 (502) times less than Mercury, which is far greater of a reduction than it's greater mass compared to Mercury.

In short, Neptune's gravity affects Earth far less than Mercury.

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