I've uploaded my normal .ico favicon but it's coming out completely wrong - any ideas why?
How did you upload the icon -- I'm guessing through the theme customization?
In that case the problem is that we force the scaling of the favicon to be 16x16, whereas your standard icon at http://startupdenmark.dk/sites/default/themes/startupdenmark/favicon.ico includes multiple sizes.
To fix the problem, simply go to Design → Files. Here you delete the current version of favicon.ico and upload your own version. When you add the favicon this way, we perform no magic on the icon at all.
(And I'll make a note for us to fix the autoscaling issue.)
I didn't .ico files could contain several favicons. I picked the smallest, renamed it favicon.ico and uploaded it but it still looks off. What am I doing wrong?
It's probably just a matter of caching in your browser since it's looking great on my end. See the attached screenshot where the icon displayed in the address bar is the cached version and the lower one is the one actually hosted on your site at the moment.
Try flushing your browser cache and then check out http://startupdenmark.23video.com/favicon.ico.