Goal Kick 2nd Touch
 

May the player that takes a goal kick be the first to touch the ball after the goal kick? (I.E. Can a player essentially dribble the ball out on a goal kick?)

No. The ball has to clear the box on the kick. No penalty if it doesn't, they just have to re-kick it. If they keep doing that, you may have a Delaying the Restart card. ;)

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Then once the ball clears the box on the goal kick, may the person that took the goal kick be the first player to touch the ball?

The answer is still no. A goal kick is considered a free kick, and like any other free kick, the kicker may only touch the ball once. Then the ball must touch or be played by another player before the original kicker can touch it again.

Law 16, third bullet point under Procedure.
"the kicker does not play the ball a second time until it has touched another player"

Yeah, sorry, I thought it was a given that the person kicking the ball cannot touch it again until another player has touched it.

Thanks everyone.

Free kicks must be touched by another player (from either team) before the player taking the free kick can touch it agan.