I was a third time applicant and am finally heading to medical school this fall. If there is a stigma, who cares? Put it out of your mind and focus on the job ahead of you. As a third time applicant, I definitely agree that the burden of proof is on us, and regardless of how many attempts you make, you must continue to show growth in your application.
Being a third time applicant definitely has its pitfalls ... self doubt, expense, and the all important element of time. Some people will probably walk away after 2 rejections. I definitely thought about throwing in the towel, but instead of walking away, I decided to hunker down and learn from all of my past mistakes (applying late, applying narrowly and to too many reach schools, not having enough clinical/shadowing experience, and not doing enough to educate myself about the problems/issues facing the medical community).
At least for me, finally making the leap from re-applicant to earning an acceptance letter took taking a brutally honest look at my application and working hard for 2 years to make it better. Look at your MCAT - can you improve it by a few points? Look at your GPA - can you add a few more tenths/hundreths of a point by taking extra classes? Revisit the essays - was something missing? Look at your EC's - are they current? Can you add more volunteering/shadowing experience to your resume? Think about your past interviews ... what was missing? Was there a perception that you don't know what you're getting into? Do you have enough drive, motivation and enthusiasm to make it through medical school?
Perhaps one of the best things that I did between the 2nd and 3rd application was making sure to take advantage of exit interview opportunities. The comments of adcom members helped me focus in on the weak parts of my application.
One thing to be wary of as a 3rd time applicant is to read each school's fine print regarding re-applicants. Some will only allow students 2 attempts, so be careful not to waste your money. Sure, being a third time applicant is definitely a difficult road to walk ... but there needn't be any stigma (at least in your own mind). Apply with the same vigor and determination of someone applying for the first time. The journey is long ... but it makes getting that first acceptance letter oh so sweet.
My favorite question is: What do you call someone who has to apply three times before getting into med school?
Doctor!
Bottom line: Just do all you can do to avoid having to make that 3rd application attempt: Apply broadly (as much as you can possibly afford ... many posters here recommend applying to 20-30 schools) and apply smart & early (make sure you can go to interview day knowing that you have done everything you could possibly imagine to improve your app). If you do happen to end up in the 3rd time around crowd ... it's no big deal. What really matters is making sure your application gets stronger with each attempt, rather than reaching a plateau.