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I see than banner far too often.
SE decided in the beginning that there should be one account per site, and not a global network account. I disagree with this choice to some extent, but I'm also not really a typical user, and the choice of separate accounts has certainly some merit. So I'm not trying to convince SE to change this, but I'd like to eliminate some of the more annoying aspects this choice inflicts on users that are active on many different SE sites.
The primary annoyance for me is that I have to log into SE sites far too often. Part of the reason is that I use SE from three different desktop computers, a tablet and a mobile phone. The other part is that SE for some reason tends to forget that I was logged in. This is certainly intentional to some part, but I suspect it also happens sometimes unintentionally. Regardless, this means that I see the banner mentioned at the beginning on every single site I visit.
What I'd like to have is the option to log into all the SE sites at once. I think this would even make sense as default behaviour, but simply having the possibility to do that would be enough for me. This wouldn't create any account on sites where I don't have one, but it should log me into all SE sites at once.
This would eliminate the major annoyance that the fragmented account system causses.