Ok, good. A thread on this. I have the exact same problem with the little zero, "0" appearing in the top left corner whenever I click on the reader. The tabs remain a the top of the page (New Post/ Reader/ Notifications/ Stats/ My Blogs/ Freshly Pressed). All of them work except when I click on the Reader Tab, which goes to this url: http://wordpress.com/#!/read/
On that tab, the page below the tabs is empty except for a small zero in the upper left hand corner.
I am doing the trouble shooting you asked for. Here are my results, as I follow your instructions.
I cleared the cache and cookies by google's instructions, which they call clearing your "browser data."
I have no addons working on this browser, and it is the latest Google Chrome version: 19.0.1084.52 m
Third party cookies are enabled. "Allow local data to be set" is selected.
I tried this reader in Firefox also, and with the same 0 result at the top of the reader page.
I have it selected to always use https.
I will now reboot, reset the modem and come back. My guess is that none of this will work and that someone at WP will have to admit it is a design flaw. I hope you prove me wrong though, because I just want it to work.