The biggest time suck in the application process is the "waiting to be verified steps" in between when you turn things in.
Even though AMCAS waited until June 21st this year to send out primaries, it is still worthwhile to finish the primary and send it in as close to June 5th (or whenever the primary is released this year) as you can (making sure you spellcheck and have others read, etc well before the deadline). This way you have as much time as possible to begin writing the secondaries (which are posted on this forum, so you don't need to wait until the school sends you the essay topics to begin writing).
Therefore, with all due respect to the posters who argue that urging people to turn things in super early is the height of SDN neuroticism (it is neurotic, but it's also one of the best ways to help yourself succeed in this process) I would urge you to have everything turned in as early in August as possible, if you're even going to wait until August. July is much, much better.
I'm sure there will be plenty of people who will reply to this and say they were average for matriculants (3.65, 32 MCAT) sent in their secondaries in aug/sept/october, were complete in sept/oct/november, and had no problems, (I am one of them...i wish I could go back and listen to my own advice!) but I would still caution you to apply as early as you can.
Turning secondaries in at the beginning of August means that the school may take a month or longer to consider your application "complete" and then send it out to be reviewed for an interview invite. Complete in September is unfortunately no longer early given 22+ MCAT exam dates and increased applicant numbers. Wait a little longer or have some kind of AMCAS verification error/LOR fiasco and you can look forward to getting caught in the slow down that happens from just before Thanksgiving until after Jan 1. At this point you are looking at waiting until Jan to get invites, Feb/March to interview.
Finally, in addition to knowing that you are done with the first step of the process and complete at your schools, applying early usually translates into earlier interviews, greater chances of gaining an acceptance from these interviews, and knowing that you have acceptances as early as possible so you can start planning your first year in medical school, whether that means saving enough $$ to move to the city in which your dream school is located or whatever. Also, you will ensure that you meet the priority deadlines for financial aid (generally March 1st for FAFSA, March 31st to have the schools' individual aps and your taxes completed).
I know this sounds like fear-mongering, but you the added peace of mind that comes from getting stuff done early is so, so worth it in this crazy process.
Good luck!