What is the time limit a new batsman has to reach the crease?

A batter (or batsman) is not timed, as far as I know, in test cricket. It may be different in some of the other versions.

If the batsman hits the ball without it being caught or running to score a run he will not be the batsman at the end of the over (six balls). His colleague will be the batsman for an over and then it will be his turn again.