Which one is the most appropriate:
- He is the bravest man to ever exist.
- He is the bravest man that has ever existed.
- He is the bravest man ever existed.
Only number 3 is ungrammatical.
For a relative clause where the relativised element is the subject, that or the relative word who/which may not be omitted.
As it is, number three gives a passive voice interpretation and since exist is an intransitive verb, this produces an undesirable result. Compare:
He is the bravest man ever interviewed by this magazine.
It could be fixed simply by removing existed
He is the bravest man ever.