What is it to Google? Apart from the obvious increase in the user base and the potential to advertise suggest websites right in the address bar, wide distribution of Chrome should somewhat increase the amount of user-generated traffic flowing into Google servers. Indeed, the default configuration of Chrome, equipped with that marvelous auto-suggestion feature, seems to start sending stuff out as soon as you type your first character into the address line.
Although the term "privacy violation" is the first one to pop out, let's keep that aside for now. The really interesting question concerns the nature of this constant influx of half-typed URLs and "search terms", annotated with timestamps and host IPs. Firstly, it most certainly contains additional value over whatever is already indexed in the web: global events, current social trends, new websites, ideas and random creative thoughts leave a mark on your address line. It therefore makes sense to look into this data and search for patterns. Secondly, the volume of this data stream is probably quite large whilst the noise is significant, hence it does not pay off to store it somewhere. And that's where we reach a nice observation: for many purposes you don't need to store this data.
If the bandwidth of the stream is constantly high, you can afford to throw it away. If at any moment you should need data, just turn on the sniffer "put your bucket in", and you'll collect megabytes of interesting stuff in a matter of seconds, if not less. A simplistic version of such kind of "stream analysis" looks as follows: you ask Google "what do people read right now", it then listens to the stream for a second and responds with something meaningful, and I believe much cooler things can be thought of. Anyway, the important point is that no "global" indexing or data collection is needed to perform this service, just a thumb on the pulse of the web.