Which teams have played the most extra-inning games over the last decade?

Wednesday's slate saw three games carry over into extra innings, with the Giants earning a win over the D-backs in 10 innings, the A's walking off the White Sox in 14 innings and the Twins notching an exciting walk-off victory in 16 innings vs. the Indians in Puerto Rico.
For fans, these extended games can be an odd experience  -- while your favorite team did not manage to win the game after nine innings, they also managed to not lose the game. Then, they get to enter the infinity and beyond known as extra innings.
We've already had a good amount of free baseball in the 2018 season -- 31 extra-inning affairs, to be exact -- but what can the last decade of extra-inning history tell us about these strange games?
Let's take a look:

Over the last ten full seasons (and the first few weeks of this season), there have been 2,137 extra-inning regular season games. No team has played in more of them than the Braves, who have played in 168 extra-inning affairs, with the Giants and Phillies coming in right behind them at 166 and 164 respectively. The Braves also boast the single season with the most extra-inning games -- they played 26 extra-inning games in 2011; a whopping 16% of their games went into extras! 
On the flip side, the Rockies have played only 110 extra-inning games over the last ten years, with the Rangers a step ahead of them at 112. The Angels played only four extra-inning games in 2016, the fewest for any team in a single season over the last decade. 
But which teams have actually excelled in extras? Once a game extends beyond the ninth, it feels like anything is possible and both teams are equally capable of pulling out a victory. Of course, that isn't really the case -- whether it's luck or some sort of difficult-to-quantify team trait, recent history shows us that some teams have performed significantly better than others in extra-inning games. In the same way that some teams play better on Fridays or when it's below 57 degrees or when the field is turf.
Why? We have no idea. But numbers are numbers, and the numbers show that the Orioles love playing past the ninth inning:

In 147 extra-inning games since the start of the 2008 season, the Orioles have won a stunning 96 of them, good for a .653% winning percentage, which is roughly a 106-win pace over a full season. Meanwhile, the Mariners have had similarly stark results in the other direction -- Seattle has won only 64 of the 152 extra-inning games they've played since 2008, a .421 winning percentage, or a 68-win pace.
Interestingly, the Mariners are also the only team left yet to play an extra-inning game in 2018 -- recent history tells us they might prefer to keep it that way.