> This is a common website which Safari has connected with many times. The error message gives no recourse. Is there a way to increase the redirection limit to allow it to open?
No. and it wouldn't help anyway.
Read the message.
The link you followed on Page A sent a redirect to another page, Page B. That other page B redirected back to the original Page A which then redirected to the other page B, which redirected back....
You have a circular loop. At no point, even if the redirect limit was set to infinity, would this ever resolve because A -> B -> A -> B ad infinitum. That's why Safari gave up.
As noted, this is usually caused by an error on the site more than anything that Safari is doing. In this case since clearing your cookies seemed to have resolved the issue, I posit that Flickr's authentication code was at fault and clearing your login (and having to log in again) resolved the issue.