I have an "innie". Strangely, although I have multiple surgical scars on my abdomen (my tummy looks a bit like a railroad map), the surgeons have always carefully bypassed my navel. I asked about this once, and the surgeon told me that many patients become greatly distressed to find themselves navel-less (belly un-buttoned?), and there are few procedures that mandate the exact trajectory of an incision, so surgeons typically work around the navel, sparing it from the knife.
In Japan, plastic surgeons can turn an "innie" into an "outie", or vice versa.
Japan is the world centre of bizarro surgeries.