The Kasparov-Karpov rivalry was exceptional. So talented and so much animus and so many games.
We needed a lot more Capa-Alekhine games though by my count they played almost 50.
McDonnell-Bourdonnais had a series of 85 games, and they were the best of their time.
I can't think of any Great vs. Great 100+ games.
Pillsbury and Lasker
The Kasparov-Karpov rivalry was exceptional. So talented and so much animus and so many games.
We needed a lot more Capa-Alekhine games though by my count they played almost 50.
McDonnell-Bourdonnais had a series of 85 games, and they were the best of their time.
I can't think of any Great vs. Great 100+ games.
I believe Karpov/Korchnoi ans Smyslov/Botvinnik were both over 100 game rivalries but none approach the rivalry of Kasparov/Karpov with over 180 games AND 5 would championship matches !
In terms of animosity rather than the number of games, Lasker/Tarrasch is hard to beat. "Für Sie, Herr Lasker, habe ich nur zwei Worte: Schach und matt!". Talk about chess as a struggle...
Karpov-Kasparov or Karpov-Korchnoi.
Death versus The Knight in the Seventh Seal.
Only one game. But there was never more at stake! 
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I had a discussion recently with a friend about great chess rivalries in top flight chess and we came up with Kasparov/Karpov as probably the greatest. They played more than 180 games against each other and were huge rivals for 2 decades. Does any other rivalry come close to theirs? I know Smyslov/Botvinnik also played more than 100 games against each other and am now curious as to how many other rivals have 100 or more games against each other? Karpov also played about 120 games against Korchnoi ! So, Karpov has 300 games just against Kasparov and Korchnoi ! I cant think of any other player that has 2 such rivals in their career, are there any ?