Well, they gain a tremendous amount of insight into your thoughts, plans, mood, etc by Chrome being your input/output mechanism for information. They generate really interesting histograms about your browsing behavior, bundle them up, and send it home to the mothership regularly. This is part of the overall strategy of knowing you, and Chrome is an edge they have over their advertising competitors by collecting this data. Google is a creepy voyeur.

I'll turn your question around a bit, though. Perhaps you step away from using Google.com, the search engine, but continue to use Chrome as your primary browser. This gives Google an intimate view of your activities; they still can build interesting trends locally/state/country/globally based in-part on your surfing behavior. In this way, I view Chrome as being just as important as Google Analytics or google.com search engine to Google's ad biz.

I personally deny Google my search traffic (I default to DDG), browser histograms (I'm on Firefox), and some of my browsing patterns on affiliates (via Privacy Badger, etc.) Obviously, they still know a ton about me by my more promiscuous past and due to my network of less privacy-conscious friends and family. But, they aren't intimately in-touch with me day-to-day and they don't know what I am thinking about up-to-the-minute like they do with their other users.