In my opinion the 3 key challenges for company like Google, which is immensely focused on innovation, solving problems related to internet and has the mission to manage world information would be...
1) Culture: With its ever expanding reach in terms of # of employees, locations, offices, teams, products and services how Google make sure to continue maintain the culture of first of all HIRING the right people and then making sure the INNOVATION doesn't die down. There are only so many smart creative people in the world or only so many can be made to adopt that thought process. DNA of Google is innovation, smart people and think BIG (10X), which I think is crux of its success.
2) Diversification: Striking the right balance between how much to diversify in product lines and how much to control. Google is moving or has moved into into robotics, android, mobile phone, laptops, cloud, computer engines, solar, energy, internet, cars, google glass, artificial intelligence, wedding planners, social media and many more. The key point is how to keep control of all these product lines and also to understand how they will help Google achieve its mission.
3) Competition: Facebook, Twitter, Microsoft, Baidu, DuckDuckGo, etc. are posing competition and coming from all directions. Google will be devise a strategy on how to maintain its lead in SEARCH space, advertising space and other revenue sources. Sundar Pichai is new head of almost all products except YouTube, so it would be interesting to see what are the next BIG things Google will come out with.