I haven't quite seen every necessary film of these collaborators to really merit this list, but nevertheless I bring to this plate my impressions thus far of these partnerships.
Jean-Luc Godard and Anna Karina - Married at the time of their film collaborations, Karina was key to the most fruitful period of Godard's career - the early to mid 60s. She was the lead in many of Godard's films during this period and she proved to adapt well to the different themes and atmospheres, without forsaking their shared interest in whimsy. She could play the philosophical hooker yearning for fulfillment in her most harrowing role as Nana in Vivre sa vie, but then be joie-de-vivre as the clucky singer in the technicolored Une femme est une femme. One film that showcases her ability to switch the nature of her character is Pierrot le fou, where her seeming childlikeness and sense of adventure mask darker, femme-fatale motives.
Woody Allen and Mia Farrow - The whole scandal aside, Allen and Farrow were quite the creative collaborators during the 80s period in which she was his muse. Woody gave Mia showcases for her sometimes underused talents and she in return is part of the reason why Allen's 80s period has so much breadth and is arguably his most consistent as a filmmaker. I think Allen distinctly understood Farrow's talents and her versatility, and so she pretty much never plays the same role twice in a film of his - whether it be as the Italian broad Tina Vitale in Broadway Danny Rose, the fragile Cecilia in The Purple Rose of Cairo, the surface-happy title character in Hannah and Her Sisters or the likeable ditz who becomes a star in Radio Days.
Martin Scorsese and Robert De Niro - Is it a coincidence that (arguably) De Niro's film roles haven't been quite as good as his glory days with Scorsese in the 70s-early 90s? There's is a partnership of an actor perfectly projecting the director's intentions onto film. The faith towards each other's works (which is crucial to the perfect partnership) can be seen in such films as Taxi Driver, The King of Comedy and De Niro's Oscar winning Raging Bull among others. Many of De Niro's characters under Scorsese are troubled, tumulting into some sort of madness, but always compelling and rendered with sensitivity.
Frank Capra and Jean Arthur - I could have gone with Capra and Stewart, but I really like the dynamics moreso between Capra and Arthur. Arthur fans would know of the often-told story that she was a bundle of nerves on set with more than a fair share of insecurities. According to Capra's son on the Mr. Smith Goes to Washington dvd commentary, Capra didn't mind the challenge that was Arthur's nerves. Indeed getting Arthur to deliver stellar performances was something of an accomplishment. Thus, a trusting director with confidence in an actor's ability can sometimes have the final say between a mediocre and good performance. I wouldn't say that Arthur didn't perform as well with other directors (she certainly delivered well for Stevens and Leisen), but that some of her performances for Capra - especially Mr. Deeds Goes to Town and Mr. Smith Goes to Washington - are at another level of maturity, assuredness and strength, with classic Arthur vulnerability thrown in.