What is the highest possible completion percentage, and how do you get over 100%?

The maximum legitimate percentage has been confirmed by Playdead to be 111. Percentages with 112 and more involves downloading someone else's game save, and no matter the reason, is against Microsoft's TOS.

Supercheats.com explains the 111 percent as follows:

Completing the game will net you a 76 percent rating on the leaderboards. You get 2 percent for each achievement, making a total of 22 percent (if you already have the achievements this is included in the bundle). You are given another 2 percent if it is a second play through, or if you are playing a chapter having already completed the game. Finally, you get 1 percent with each secret egg.

As you mathematicians will have realised, the total adds up to 100 percent not including the secret eggs. That means that if you have already completed the game with all achievements, you are guaranteed to get 100 percent on your next playthrough. The secret eggs add up to 11 percent, meaning the best you can get on the leaderboards is 111 percent!

You may also want to check this forum thread for some details about how they get 112 or more.