I think many experts in stack exchange have knowledge in multiple discipline, and they would like to share their knowledge across multiple domains. It is not difficult for them to have knowledge in more than 7 domains. Please make the combined flair shows more than 7 icons to encourage expert supports other satellite sites.
This isn't really needed. The whole point of the flair is to highlight your activity on the Stack Exchange network. Seven sites does that perfectly well. Adding more logos to the flair image doesn't achieve anything because those logos don't actually link to their separate sites. They all link to a single page, which has an accounts tab, which will list out all sites you're using and your reputation under each one. It's simple enough for a user to access that information and access each specific profile without the need for overwhelming them with icons on a flair image (which they may not even know the meaning of).
If you're so worried about showing off every site, try generating your own image. Or, you could even dedicate an entire page on your personal site (like I did) to explaining each site and what you get from participating in it.
I currently have 18 qualifying sites, but they're not all equally important. Browsing some random network profiles, I think that's not uncommon -- people have sites they're active on, sites they dabble in, sites they were active on in the past, maybe sites where they answered one hot question and got a pile of rep... so just showing all those sites doesn't really tell people about you. Further, sometimes that lower rep on a niche site means more than higher rep on a more-popular site.
Perhaps, instead, the needs of polymaths across the network would be better met by letting people choose the seven sites to go into the flair. If you're really proud of your #8 site and you haven't visited your #4 site in months anyway, go ahead and change the lineup. If your top sites are somewhat redundant (maybe you don't need both Ask Ubuntu and Unix & Linux in your flair, for example), go ahead and make room for something that shows a different slice of your skills.
This would require an editing UI and might not be worth it, but I think you'd be better served by pursuing this code change instead of a "show everything" code change.
I agree with the OP. Refraining from adding an extra row for more site logos makes Stack Exchange (SE) biased towards (niche) experts versus generalists. Also, it renders the SE gamification less appealing towards generalists.
This artificial limitation is not very smart move for a company like SE, as many recruiters are actually scouting for a T-shaped skill profile.
So SE, don't be anal about this and add this little bit of flair™ code; if only to prevent Leonardo da Vinci from turning in his grave!