The phrase "obvious exception" means that there is a clear and easily noticeable situation or example that does not fit the general rule or pattern. Full definition
With the rather obvious exception of the idiots who decided he should remain in charge, he's blown his credibility with most of the rest of the planet. (co2insanity.com)
There is a very strong belief that, as far as possible, and with obvious exceptions, people should have «paid into the system» before taking anything out in benefits. (totalpolitics.com)
We have a relatively healthy squad (obvious exceptions Sanchez, Debuchy, Arteta). (justarsenal.com)