This started off as a comment on Snappie's answer, but it's getting a bit long...
Assume all data is encrypted using SSL
But not all data is encrypted!
(in addition to IP address, and the amount of data being passed in either direction...)
They can also see the IP names of all the resources you access (including web) from your DNS activity (not encrypted). If you are using a proxy, then this is also exposed in the proxy logs.
They can see what type of network interface you have (from the MAC address).
They can see when your computer connects to the network (Wifi connect, DHCP).
From passive sniffing of encrypted traffic they could probably guess what family of OS you are using.
I believe Chrome can be quite chatty as it tries to work out if something is a URL or a search term and tries to diagnose what network connectivity it has. I would presume that this could be exploited to determine if you are using Chrome, although its not something I know a great deal about.
If, at any time, your browser does make a non-encrypted request, then this will reveal a lot of details about your browser and OS. Depending on the service, it may also reveal a lot about your affiliation to the site.
...and this is just from passive data collection. If they choose to start sending data at your device, it will reveal a lot more.