I'd say it's between Bayern and Barca. Bayern have the edge historically, but Barca had a greater influence on the game. Some of which you could argue was really Ajax's influence through Michels and Cruyff, but then again they also took their ideas from somewhere else.
Barca have had bigger names. Maradona before they even won the European Cup, then in the 90s Romario, Rivaldo etc., R9's best ever club season, then Ronaldinho was the biggest star in football for most of the 00s, bigger than R9 and Zidane, then Messi came.
Spain won 08-10-12 with a core of Barca players, playing to the strengths of the Barca players. Pep's Barca made everyone else in the league adapt, and La Liga became the best league in the world. BdO podium of 3 Barca players. Pretty much every top manager in the world right now citing Pep's Barca as an inspiration.
Great story too: in 2008, Mourinho was the hottest manager in the world. He had broken SAF and Wenger's dominance in the EPL, and broken records. He had worked for Barcelona. Him getting the job looked inevitable, and everyone was scared of how good Barca would become. Instead Cruyff told Laporta no, hire the B team manager. So everyone thought huh, so they're going through a rebuilding process after two trophyless seasons, they didn't go for a big name. But they instantly assembled what most people consider to be the greatest team of all time. Won the sextuple and a billion trophies, but more importantly, changed the way football was played.
If you watch a Barca, Bayern, Milan, whatever game from before Pep's Barca, you'll notice how much open space there was on the pitch. We think of the defensive Italian teams from way back, but nobody really defended as a unit. Few years later, every team attacked and defended together. No more massive gaps, no more idle time for one compartment while the ball's on the other side of the pitch.
Teams immediately tried replicating what Barca was doing. Chelsea bought Oriol Romeu hoping he'd be the next Busquets. Wenger saw it coming, he talked about it, and changed his French-based Arsenal with big, physical players to a more Spanish-style squad, based around Cesc, Cazorla, Arteta or small, technical players in general, like Wilshere and Nasri. Everyone wanted La Masia players, or Barca-type players.
And at the beginning, most copycat attempts failed. Because they all thought "alright, we get small, technical players and we play a possession game" was enough. Top 5 league managers tried copying Barcelona but couldn't, because they couldn't actually understand the tactics behind Barca's game. Instead, the managers that did truly understand the tactics, perfected bus-parking. Doesn't matter whether we see bus-parking as a bad thing or not, defensive tactics were forced to evolve massively as a reaction to Barcelona's game. We've never seen anything like it happen in reaction to SAF, Jupp or Ancelotti's football.
Bayern may have been the more consistent club, but they've never reached Barca's peaks.