I haven't checked this guess, and have no immediate plans to do so, but I feel reasonably comfortable in guessing that the A's franchise has, if not the highest statistical variance in seasonal winning percentage among the original 16 teams, at least a top three figure. Going back to their days in Philadelphia, they've effectively alternated between periods of excellence and dreadfulness, with very little time spent being mediocre. The reason is simple: the team has often been run by extremely smart baseball men (Connie Mack, Charlie Finley, Billy Beane), but they have almost never made any money, and so they can't (or at least don't) keep their great teams together for long.
We'll get to the Philly A's later, but in Oakland, there have been three or four major competitive periods for the team: the early '70s, when they won three consecutive World Series, the late '80s, when they won three consecutive AL pennants, and the Beane era (which you can take as one decade-plus competitive period, or split into roughly 2000-06 and 2012-14 periods of strong contention), in which they've made the playoffs eight times and won zero ALCS games.
Surprisingly, the best option for this entry comes from the team whose (excrement) doesn't work in the playoffs. It's 2003 ALDS Game 1: A's 5, Red Sox 4 (12). Since it was the first game of the series, both teams had their aces going - which meant the underrated Tim Hudson for the Oakland, and the much-better-than-Tim-Hudson(-and-most-other-pitchers) Pedro Martinez for Boston.
Hudson faced four batters in the first inning. He retired Johnny Damon, Nomar Garciaparra, and Manny Ramirez - but Todd Walker hit a two-out solo homer to open the scoring. Pedro, being Pedro, retired the side in order in the first. Each pitcher allowed a single in the second; Kevin Millar was thrown out trying for second on a pitch that wasn't wild enough, and Jose Guillen was left on. Boston loaded the bases on the third on singles from Jason Varitek, Damon, and Walker, but Manny grounded out to leave all three runners on, and the A's took the lead in the bottom of the inning. Chris Singleton started the rally with a one-out double, Mark Ellis walked, and Erubiel Durazo doubled both runners home. One out later, Miguel Tejada singled to score Durazo, although Tejada was thrown out in a rundown between first and second to end the inning.
Neither Hudson nor Martinez allowed a hit in the fourth (though Pedro walked Ramon Hernandez). Boston halved their deficit on a Varitek homer in the fifth, and Hudson then allowed two-out singles to Garciaparra and Walker before Ramirez grounded out to strand them. Ellis singled and moved to third on an errant pickoff throw in the bottom of the fifth, but then was thrown out at home on a grounder back to the mound. Millar's single made him the only runner on either team to reach in the sixth.
Varitek opened the seventh with a walk, but Damon erased him with a double play. However, Garciaparra then singled, which brought Walker to the plate, and given Walker's success against Hudson on the day, Ken Macha decided to pull his starter for lefty Ricardo Rincon. Naturally, Walker hit Rincon's third pitch over the wall for a go-ahead two-run homer.
Oakland threatened in the bottom of the seventh. Hernandez led off with a single; Jermaine Dye hit into a force, but took second when Walker's attempt to complete the double play went awry. After the second out, Ellis and Durazo walked to load the bases, but Eric Chavez fouled out to strand all three runners. The Red Sox mounted a similar rally in the eighth; David Ortiz led off with a walk, and Bill Mueller's one-out double moved him to third and chased Rincon. Chad Bradford relieved and escaped with a strikeout, an intentional walk, and a forceout.
Mike Timlin worked a perfect bottom of the eighth, and Keith Foulke returned the favor in the ninth. And then the Red Sox turned the game over to Byung-Hyun Kim - because when has that ever gone badly in the postseason? Pinch hitter Billy McMillon drew a one-out walk, and Singleton was hit by a pitch. Ellis struck out, putting the A's one out away from defeat, and Alan Embree was brought in for Kim. He allowed a single to Durazo, which scored pinch runner Eric Byrnes from second with the tying run. Chavez then grounded out, sending the game to extras.
Foulke was perfect in the tenth, while Scott Williamson walked Scott Hatteberg, but then escaped the inning on a strikeout looking/runner out advancing double play, which is an unusual combo. A two-out Damon walk, a steal, and an intentional pass to Nomar put the Sox in business in the eleventh, but Walker had been pulled for defensive sub Damian Jackson back when Boston was ahead, and Jackson fanned to strand both runners. Derek Lowe came in for the home eleventh and allowed a walk and a steal, but nothing else.
Rich Harden walked Ramirez and wild pitched him to second to open the twelfth; he would intentionally walk Mueller two outs later, but retired the other three Red Sox he faced in the inning. Durazo also drew a leadoff walk in the bottom of the inning; Chavez hit into a force, then moved to second on a groundout. Hatteberg walked, with Chavez stealing third on ball four, and Terrence Long was intentionally walked to load the bases.
Up next was Ramon Hernandez. Even by the low standards of catchers, Hernandez was not a fast man. He spent 15 seasons in the majors, and ended up with eight triples and nine stolen bases in his career. Baseball-Reference credits him with costing his teams 29 runs via baserunning alone over the course of his time in the majors. And given that he played for the Moneyball A's, one of the most famously station-to-station teams in recent memory, the last thing anyone on Earth would have reasonably expect him to do was bunt.
But bunt he did, a high, slow bouncer to the third base side. The Red Sox, understandably, had the infield fairly deep, and Mueller had no play by the time he fielded the ball. The A's had walked off with the game on a twelfth-inning bunt single, after trailing with two outs in the ninth.
This game is utterly fabulous; it's one of the 15 most-dramatic playoff games ever, per WPL. If the A's had won the series - or, even better, won more than one series this year - it would be the stuff of legend. Sadly, they turned in their traditional collapse from ahead 2-0 - but at least this time, it was a highly interesting one, as WPL grades the 2003 Sox-A's ALDS as one of the top three best-of-5 series ever played, and that's without accounting for the walkoff bunt (or the baserunning shenanigans in the later games of the series).
The Red Sox, meanwhile, would go on to lose to the Yankees in a not-quite-as-good-overall but more famous series; they would finally end their title drought next year, thanks in part to the Oakland pitcher who shut them down for three hitless relief innings in this game, Keith Foulke.
The A's have won 58 postseason games since moving to Oakland - and only four of them are in the top 200 all-time (you'd expect roughly twice that). The other three still make up a solid honorable mention section: 1973 World Series Game 3, a Catfish Hunter-Tom Seaver duel that the A's won in 11 after rallying to tie in the eighth, 1972 ALCS Game 1, another Hunter start against the opposing ace (Detroit's Mickey Lolich this time) which saw the Tigers take a 2-1 lead in the eleventh only to have Oakland come back with two in the bottom of the inning (the second of which scored on a walkoff throwing error), and 1988 ALCS Game 2, which was scoreless until the sixth, then the Red Sox scored twice, then the A's immediately tied it on a Jose Canseco homer in the seventh and pulled ahead later in the inning, then Rich Gedman of all people homered to tie it in the bottom of the seventh, and the A's broke the tie in the ninth.
Remember, of course, that we're not done with the A's quite yet. And if you think the honorable mention pickings are slim for the Oakland version, just wait until we head back to Philly.