How to increase Chrome's connections-per-server?

I'm sorry but it would appear to be a hard limit according to http://code.google.com/p/chromium/issues/detail?id=12066

Actually you're limited per profile, not per browser.

All you have to do is to create several profiles, from the settings page. I have two for Facebook accounts, one for my sister and one for work. In total I could be able to download 18 files concurrently and all I have to do is to switch or bring up alongside a new Chrome profile in its own window from the top-left corner icon, viewable only when there are more than one profiles. And it's pretty fast as well, reuses Chrome apps (flash etc) and main exe so it's not like it's a whole new browser launched with it's own resources.

Unfortunately there's no other option...

As long as your connections are on HTTP2 (and not HTTP 1.1), then you're not affected by this limit.