are there any medical school

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cause this is the thing - medical career is FULL of tests, board exams, etc. I personally, would advise you to take MCAT to see if you are comfortable with that. I knew a guy who went to a school that didnt require MCAT (a llong time ago), and he barely passed step 1, and with step 2 - he gave up after failing it 5 times.....
 
Most BA/MD programs will also require a minimum MCAT score to secure the provisional acceptance to the linked MD program. It is not a perfect test by any means, but neither are the boards. Most (if not all) medical schools use the test as one indicator to gauge whether a student will be more likely to pass their boards later on. I would take the exam when you are ready.
 
Not that I know of
I was browsing the ChooseDO Explorer and I saw LECOM as one. Probably others.

No idea if there any people who are actually successful without one though, but they did list it:

The College will consider the applicant’s most recent Medical College Admission Test (MCAT) scores. MCAT scores taken within the past three years are acceptable. A competitive MCAT score is in the 50 percentile or higher. In lieu of MCAT Scores, LECOM may use an Academic Index Score that uses undergraduate and graduate GPAs in a formula calculation with ACT and/or SAT Critical Reading and Math scores. Use the form below to calculate your Academic Index Score.
 
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