This is a common question, and here's exactly what you need to know:
PREFACE: You must apply by mid June. June 1st or June 4th does not actually make any difference as some students believe it does. However, applying within the first 1-2 weeks of June is important for all rolling admissions schools (b/c AMCAS becomes a log-jam about late June/July and you end up way behind a long list of other applicants). After that, it just gets harder and harder and you have to depend on some luck and being somewhat of the exception. Why do that to yourself? Just apply before June 15th and give yourself the best chances possible.
1) If you want to know your MCAT score BEFORE you submit AMCAS, you've got to take it no later than mid May.
2) If you are willing to submit AMCAS in early-to-mid June, WITHOUT knowing your MCAT score, then you can test as late as the last week of July. This is b/c AMCAS will actually be processed without your score and your score will add to your application automatically 30 days after you take it. Assuming letters of rec, transcripts, etc., are all in place, you'll be ready to review about 30 days after your exam date.
If you submit AMCAS w/o a sore in mid June, you could maybe push it to a mid August exam, but then your file won't be ready for review until after the ADCOMS have already been meeting for a few weeks. By late September or early October you are definitely in the late applicant group and you could be waiting on interviews until late winter or next spring = NOT GOOD.