How many "times" can someone apply to medical school if they don't get in the first time?

You can reapply as many times as you like, but there is one major catch.  The medical schools are unlikely to consider your application again unless something on the application has changed.  It is your job to provide this change and point it out when reapplying.

You can apply twice and more than that they will close. But you can try to other University.

You can apply as many times as you want. I applied 3  times before I was accepted. After undergrad I had average grades and an average MCAT score and was rejected. I then retook the MCAT and did better, but still had average grades and was rejected again. Then I did a post-bac program and improved my grades and was finally accepted. So, if you apply and are rejected ask the dean of admissions which part of your application was weak and do something to improve it. Things that look good are GPA >3.5 minimum, MCAT >27, and strong research and/or volunteer activities.

Some schools do have a limit (usually two times) while other schools have no limits.  If there is a limit, the school's web site will usually state it.

The three critical areas being assessed on your application are your overall undergraduate GPA, your GPA in the prereq courses and your MCAT scores.  If you don't meet the school's minimum cut-off in these area they won't bother reading the rest of your application.

People who have average GPAs and scores have been admitted instead of people with stellar GPAs and scores.  Presumably this is due to an outstanding personal statement and interview.  There certainly are other factors, but it's impossible to quantify all of the variables.  But your best chance of being accepted lies in doing the best you can, even if it means repeating courses to improve the GPA, which is very common among medical school applicants (and not held against you).