Why does God not show Himself beyond reasonable doubt? In this answer I commented that Christians believe that "God does not show himself as this would take away free will", to which I was answered "The only reason he would not show himself is that it would utterly destroy us to behold him."

Which one is correct (if any)?

Examples of proof without doubt: 


He could encode the Bible in the digits of π (making it a statistical, logical and mathematical miracle).
Put His "trademark" in the rules of logic, or in our DNA. 
Write Schroendiger's law in the Bible. 
Make an exact, impossible to forge, prediction or series of predictions, which are then correct, but impossible to make for a non omniscient human.
Because God decided faith was an absolute requirement in His kingdom:


  But without faith it is impossible to please him: for he that cometh to God must believe that he is Hebrews 11:6 KJV


Actually, Jesus is apparently frustrated with those who desire the type of "sign" you suggest:


  And he sighed deeply in his spirit, and saith, Why doth this generation seek after a sign? verily I say unto you, There shall no sign be given unto this generation. Mark 8:12 KJV