From what I've seen, anything later than late July is considered late. To make life easier, get everything done as early as possible. Especially Letters, and transcripts.
This is exactly the kind of sdn fear mongering that drives so many people to utter neuroticism, and it's not supported by the evidence, either the direct reports of adcom members who post here, AMCAS's own reports, or even the SDN verification thread.
Last year, in the midst of all the delays in processing, the AAMC released an announcement in the last week of June indicating that roughly 1/3 of the eventual final applicant pool had submitted their primaries (they released the raw number that I'm too lazy to look up now, but it ended up being about 1/3 of the applicant pool). People who submitted in the last week of June were verified in early/mid August.
What does this mean? It means that the median completion time falls sometime in August, maybe Sept depending on how much AMCAS drags their feet verifying.
Additional important signposts are that AMCAS doesn't release ANY applications until the end of June, and many schools don't open up their secondaries until sometime in July anyway.
So you can quibble with the exact definition of 'early', but given that you can be complete in mid August and still beat the
majority of the applicant pool to the punch, later than late July is not 'late' to anyone but neurotic premeds on SDN. Everyone is so focused on early vs. late that they forget it's possible to be neither.
Important to note that everything I've said above refers to completion date (primary verified, secondary submitted, MCAT scores in, etc). If you submit your AMCAS later than late June/first week of July you're probably in for a bad time.